The document contains solutions and answers to exercises on propositional logic. It includes determining which statements are propositions, writing negations of statements, using truth tables to evaluate compound statements, and determining logical equivalences. Key examples show that (p∧q) is logically equivalent to p⟹q, and ((p⟹q)∧(q⟹r))⟹(p⟹r) is a tautology. The document also evaluates conditional statements based on assigned truth values.
Logic and Critical Thinking (Final)_281019125429 (1).pdfHabibBeshir
This chapter introduces philosophy by discussing its meaning, nature, and importance. Philosophy is defined etymologically as "love of wisdom" and refers to developing critical thinking habits and continuously seeking truth. The chapter outlines some key concepts in philosophy like its universal nature makes it difficult to define by subject matter alone. It also notes that the best way to understand philosophy is to actively engage in philosophical thinking and discussion. The chapter aims to provide foundational knowledge about philosophy as a discipline and rationale for its study.
This document provides an overview of the anthropology course Anth 1012 at Mekelle University. It defines anthropology as the study of humanity, including our origins, development, and cultural variations throughout history and around the world. Anthropology analyzes both biological and cultural aspects of humans. The document traces the historical development of anthropology from its roots in ancient Greek philosophy to emerging as an academic discipline in the 19th century. It describes the broad scope and unique features of anthropology, including its holistic and relativistic approach, comparative perspective, and emphasis on qualitative research methods like ethnography. Some common misconceptions about anthropology are addressed, and the relationships and contributions of anthropology to other disciplines are discussed.
This document provides an overview of the Logic and Critical Thinking course. It discusses the course content which includes chapters on logic, philosophy, critical thinking, and logical reasoning. The first chapter defines logic as a field of study and as an instrument to evaluate arguments. Philosophy is defined as the love of wisdom which deals with fundamental questions. The major fields of philosophy - metaphysics, epistemology, axiology, and logic - are also outlined. The chapter objectives are to understand the meaning and importance of philosophy. Overall, the summary provides a high-level view of the topics and chapters covered in the Logic and Critical Thinking course.
This document discusses human evolution and cultural development in Ethiopia and the Horn of Africa based on archaeological and fossil evidence. It notes that:
1) Early human ancestors like Ardipithecus, Australopithecus and various species of Homo emerged in the region between 4-2 million years ago based on fossil discoveries.
2) Stone tool technologies evolved from Oldowan to Acheulean to Sangoan modes associated with these early humans.
3) The Neolithic revolution began around 10,000 years ago when humans transitioned to sedentary agriculture and animal domestication, evidenced by sites containing crops, tools and domesticated animal remains.
4) The region is ethnically
History of Ethiopia and the Horn Common Course (2).pptxGalassaAbdi
This document provides an overview of a university course on the history of Ethiopia and the Horn of Africa. The course is designed as a common course for Ethiopian students and covers the region's history from ancient times to 1995. It is divided into seven units that examine major social, cultural, economic and political developments. The course objectives are to introduce students to the diverse histories of the region and how interactions between peoples shaped its development. Specific topics that will be covered include human evolution, ancient states, religious processes, and internal and external relations from the 19th to 20th centuries.
The document summarizes the emergence and development of early states in Ethiopia and the Horn of Africa from ancient times until the 13th century CE. It discusses:
- The earliest recorded state of Punt located in northern Somalia or northern Ethiopia, known from Egyptian texts between 2500-1500 BCE.
- Other early cultural centers that emerged like Da'amat and Yeha in northern Ethiopia and Eritrea between 1000-500 BCE.
- The rise of the powerful Aksumite state between 200 BCE-700 CE, which dominated trade routes in the Red Sea region and had territories extending across modern-day Ethiopia, Eritrea, Sudan, and South Arabia.
- The Zagwe
History is the systematic study of past events through organized knowledge. The purpose is not just to list events but to find patterns and meaning. Historians study surviving records to write histories and interpret the past. Studying history helps understand the present, develop a sense of identity, and provides context for other disciplines. It also teaches critical thinking skills. Historians rely on primary sources like documents and artifacts as well as secondary sources like histories to research and interpret the past. The historiography of Ethiopia and the Horn has developed over time from early accounts to modern historical studies using a variety of written and oral sources.
This document discusses national interests and foreign policy. It defines national interest as the values, goals and objectives a country aims to achieve in international relations. National interests drive a country's foreign policy. The document also outlines different criteria for defining national interests, including operational philosophy, ideological, moral/legal, pragmatic, and foreign dependency criteria. It then discusses foreign policy objectives, including short-term goals of security, middle-term goals of economic welfare, and long-term visions to restructure the international system. The document also examines patterns of foreign policy behavior and dimensions for analyzing it such as alignment, scope and mode of operation.
Logic and Critical Thinking (Final)_281019125429 (1).pdfHabibBeshir
This chapter introduces philosophy by discussing its meaning, nature, and importance. Philosophy is defined etymologically as "love of wisdom" and refers to developing critical thinking habits and continuously seeking truth. The chapter outlines some key concepts in philosophy like its universal nature makes it difficult to define by subject matter alone. It also notes that the best way to understand philosophy is to actively engage in philosophical thinking and discussion. The chapter aims to provide foundational knowledge about philosophy as a discipline and rationale for its study.
This document provides an overview of the anthropology course Anth 1012 at Mekelle University. It defines anthropology as the study of humanity, including our origins, development, and cultural variations throughout history and around the world. Anthropology analyzes both biological and cultural aspects of humans. The document traces the historical development of anthropology from its roots in ancient Greek philosophy to emerging as an academic discipline in the 19th century. It describes the broad scope and unique features of anthropology, including its holistic and relativistic approach, comparative perspective, and emphasis on qualitative research methods like ethnography. Some common misconceptions about anthropology are addressed, and the relationships and contributions of anthropology to other disciplines are discussed.
This document provides an overview of the Logic and Critical Thinking course. It discusses the course content which includes chapters on logic, philosophy, critical thinking, and logical reasoning. The first chapter defines logic as a field of study and as an instrument to evaluate arguments. Philosophy is defined as the love of wisdom which deals with fundamental questions. The major fields of philosophy - metaphysics, epistemology, axiology, and logic - are also outlined. The chapter objectives are to understand the meaning and importance of philosophy. Overall, the summary provides a high-level view of the topics and chapters covered in the Logic and Critical Thinking course.
This document discusses human evolution and cultural development in Ethiopia and the Horn of Africa based on archaeological and fossil evidence. It notes that:
1) Early human ancestors like Ardipithecus, Australopithecus and various species of Homo emerged in the region between 4-2 million years ago based on fossil discoveries.
2) Stone tool technologies evolved from Oldowan to Acheulean to Sangoan modes associated with these early humans.
3) The Neolithic revolution began around 10,000 years ago when humans transitioned to sedentary agriculture and animal domestication, evidenced by sites containing crops, tools and domesticated animal remains.
4) The region is ethnically
History of Ethiopia and the Horn Common Course (2).pptxGalassaAbdi
This document provides an overview of a university course on the history of Ethiopia and the Horn of Africa. The course is designed as a common course for Ethiopian students and covers the region's history from ancient times to 1995. It is divided into seven units that examine major social, cultural, economic and political developments. The course objectives are to introduce students to the diverse histories of the region and how interactions between peoples shaped its development. Specific topics that will be covered include human evolution, ancient states, religious processes, and internal and external relations from the 19th to 20th centuries.
The document summarizes the emergence and development of early states in Ethiopia and the Horn of Africa from ancient times until the 13th century CE. It discusses:
- The earliest recorded state of Punt located in northern Somalia or northern Ethiopia, known from Egyptian texts between 2500-1500 BCE.
- Other early cultural centers that emerged like Da'amat and Yeha in northern Ethiopia and Eritrea between 1000-500 BCE.
- The rise of the powerful Aksumite state between 200 BCE-700 CE, which dominated trade routes in the Red Sea region and had territories extending across modern-day Ethiopia, Eritrea, Sudan, and South Arabia.
- The Zagwe
History is the systematic study of past events through organized knowledge. The purpose is not just to list events but to find patterns and meaning. Historians study surviving records to write histories and interpret the past. Studying history helps understand the present, develop a sense of identity, and provides context for other disciplines. It also teaches critical thinking skills. Historians rely on primary sources like documents and artifacts as well as secondary sources like histories to research and interpret the past. The historiography of Ethiopia and the Horn has developed over time from early accounts to modern historical studies using a variety of written and oral sources.
This document discusses national interests and foreign policy. It defines national interest as the values, goals and objectives a country aims to achieve in international relations. National interests drive a country's foreign policy. The document also outlines different criteria for defining national interests, including operational philosophy, ideological, moral/legal, pragmatic, and foreign dependency criteria. It then discusses foreign policy objectives, including short-term goals of security, middle-term goals of economic welfare, and long-term visions to restructure the international system. The document also examines patterns of foreign policy behavior and dimensions for analyzing it such as alignment, scope and mode of operation.
This document provides an overview of international political economy (IPE). It discusses the meaning and nature of IPE, focusing on the political and economic dimensions. It outlines three major theoretical perspectives on IPE: mercantilism/economic nationalism, liberalism, and Marxism. Differences in national political economy systems are examined through case studies of the US, Japan, and Germany. Finally, core issues in IPE governance are discussed, specifically international trade and the role of the World Trade Organization.
The systematic study of science and religion was started in 1960s. Science and religion had been defacto Western science and Christianity for the past fifty years. One way to distinguish between science and religion is that Science concerns the natural world, whereas Religion concerns both the natural and the supernatural world. Barbour’s 4 models of science and religion interactions are Conflict, Independence, Dialogue, and Integration Models. Some philosophers suggest that Christianity was instrumental in catalyzing scientific revolution. Contemporary lack of scientific prominence is remarkable in the Islamic World. The two views of divine action are general divine action and special divine action. Evolutionary ethics & Implications of the cognitive science of religion are areas of increasing interest in science and religion.
This document provides an overview of an inclusive education course developed by several universities in Ethiopia. The course is intended to be required for all undergraduate students. It aims to teach students how to create inclusive environments and provide appropriate support to people with disabilities and vulnerabilities. The course objectives, chapter outlines, and learning outcomes are described. The first chapter focuses on defining key terms, describing various types and causes of disabilities, the history of inclusion, models of disability, and the impact of attitudes on inclusion.
This document discusses the electromagnetic spectrum and properties of light. It describes how light exhibits both wave-like and particle-like properties. The wave properties of light include frequency, wavelength, speed and amplitude. The particle properties include photons and the photoelectric effect. The document also covers the Bohr model of the hydrogen atom and how it led to the development of quantum theory, which explained atomic spectra and the dual wave-particle nature of matter and energy.
This document provides an overview of basic concepts in logic, including:
- Logic is defined as the science of reasoning, concerned with determining correct vs incorrect reasoning.
- Reasoning involves making inferences by drawing conclusions from premises. Arguments consist of statements where one is the conclusion and the others are premises intended to support the conclusion.
- Logic divides into deductive logic and inductive logic. Deductive logic examines necessarily truth-preserving arguments, while inductive logic examines arguments where the premises make the conclusion probable but not certain.
Ethiopian Law of Evidence Lecture Notes pptgetabelete
This document provides a summary of Berhe Gessesew's lecture on Chapter Two of Evidence Law regarding facts that may be proved other than by evidence. The summary is as follows:
1. Generally, parties have the burden of proving allegations, but some allegations do not require proof if admitted by the opposing party, presumed by the judge, or subject to judicial notice.
2. Admitted facts do not require proof because parties are assumed to know their own facts and courts aim to resolve disputes rather than prove undisputed facts. Admissions can be formal, informal, judicial or extra-judicial. More weight is given to formal admissions made to those with authority.
3. Presumptions shift or fix the
This document provides an introduction to logic. It outlines the objectives of studying logic as sharpening intellect, developing learning ability, strengthening understanding, and promoting clear thinking. Key benefits include supporting reasoning powers, distinguishing good from bad arguments, and learning principles of clear thinking. Logic is defined as the study of correct versus incorrect reasoning. Important concepts discussed include premises, propositions, arguments, sound versus unsound arguments, and laws of logic. The overall aim is to learn how to evaluate arguments.
CHAPTER ONE & TWO LOGIC AND PHILOSOPHY.pptxBarentuShemsu
This document provides an introduction to philosophy by outlining some of its key concepts and fields. It begins by defining philosophy as the love of wisdom and noting that philosophy deals primarily with fundamental issues rather than having a single subject matter. The document then outlines some of philosophy's major fields, including metaphysics, epistemology, axiology, and logic. For each field, it provides brief definitions and examples of the types of questions addressed. The document emphasizes that philosophy is an activity that encourages critical examination and reflection on life and reality.
This document discusses different research philosophies and approaches. It defines key characteristics of research paradigms such as ontology, epistemology, and axiology. It presents the "research onion" as a model of the layers within research philosophy. The main research philosophies discussed are positivism, realism, interpretivism, and pragmatism. Positivism is defined as trying to uncover objective truths about how the world works through logical deduction and empirical observation in order to predict and control outcomes.
Chapter One & Two.pptx introduction to emerging TechnologygadisaAdamu
This document provides an overview of emerging technologies and data science. It discusses key concepts like invention, innovation, and technology. It describes the four industrial revolutions and how they transformed society through innovations like the steam engine. Emerging technologies discussed include artificial intelligence, blockchain, and robotics. The role of data in powering emerging technologies is also examined. The document then shifts to discussing data science, defining data, information, and knowledge. It outlines the data processing cycle and common data types used in computer science.
This document provides an overview of philosophy and the philosophy of the human person. It defines philosophy as the systematic study of truth and the principles of beings through reason and faith. The document discusses the etymology and definitions of philosophy. It also outlines different levels and branches of wisdom and philosophy. In particular, it distinguishes between natural wisdom gained through reason and supernatural wisdom gained through faith. The document contrasts philosophy and theology, and discusses the importance and theoretical and practical branches of philosophy, including metaphysics, ethics, and aesthetics.
The document provides an overview of topics covered in the first week of a symbolic logic course, including:
1) The basic components of an argument such as premises, conclusions, and deductive validity.
2) Ways an argument can be weak, including having false premises or premises that don't support the conclusion.
3) Logical concepts like necessity, possibility, consistency, and logical equivalence.
4) The difference between formal languages like sentential logic and natural languages.
This document provides an overview of Western philosophy from ancient to medieval periods. It discusses some of the major philosophers from each era and their contributions. The ancient Greek philosophers like Socrates, Plato and Aristotle established the foundations of Western thought. Medieval philosophy integrated Greek rationalism with Christian theology. Major philosophers included Augustine of Hippo who wrote extensively on theology and philosophy. The document outlines some key characteristics of medieval philosophy like theocentrism, theodicy, and emphasis on God and faith.
Logic is the study of reasoning and correct thinking. It involves analyzing concepts, establishing general laws of
truth, and determining valid forms of argument. Logic is applicable to all fields as it provides standards for
consistent and evidence-based reasoning. It has wide scope and helps with social studies, engineering, mathematics,
science, and computer programming through modeling reality, simplifying complex problems, and representing
information processing in a logical way. Studying logic is important as it helps develop critical thinking skills
needed to make rational decisions, adapt to new situations, and form justifiable beliefs.
This document provides an overview of the course "Logic and Critical Thinking". It discusses the following key points:
1. The course covers 6 chapters, including introductions to logic, basic logic concepts, critical thinking, logical reasoning and fallacies, categorical propositions.
2. Chapter 1 defines philosophy as the love of wisdom and discusses its major fields including metaphysics, epistemology, axiology, and logic. It emphasizes that philosophy questions apparent truths.
3. Logic is the study of arguments and their structures. An argument consists of premises that provide support for a conclusion. Identifying premises and conclusions is important for evaluating arguments.
This document provides an introduction and overview of philosophy. It defines philosophy as the systematic study of fundamental human knowledge and the pursuit of wisdom. The document traces the origin of the term "philosophy" to ancient Greek roots meaning "love of wisdom". It outlines the main goals of philosophy as discovering the nature of truth and knowledge. The document also describes the scope and key branches of philosophy, and emphasizes philosophy's importance in clarifying beliefs, stimulating thinking, and developing analytical abilities.
The document discusses ancient Greek philosophers who lived before Socrates, known as the Pre-Socratics. Three philosophers from Miletus - Thales, Anaximander, and Anaximenes - believed the basic substance of the universe was water, infinity, and air respectively. Parmenides argued that change and motion are illusions and all is permanent. Heraclitus believed all is in flux. Empedocles said nature's source cannot be a single element. Anaxagoras believed seeds were ordered by intelligence. Pythagoras and his followers studied mathematics and believed the universe followed musical and numerical laws, discovering the Pythagorean theorem relating triangle side lengths.
Plato and the Pre-Socratic philosophers attempted to explain reality through reason alone, without reference to myth or religion. They debated questions about the fundamental nature of reality and whether it is one or many. Plato believed in eternal "Forms" that were the perfect essence of things in the world, which were imperfect copies. Aristotle rejected Plato's theory of Forms and argued reality is composed of both matter and form, with accidents being non-essential features and essences being defining characteristics. This document provides an overview of the major philosophers from Thales to Aristotle and some of the key debates around the nature of reality, knowledge, and ethics.
The document discusses the basics of logic including propositions, truth tables, and logical connectives. It defines a proposition as a statement that is either true or false. Compound propositions can be formed using logical connectives like AND, OR, XOR, NAND, and NOR. Truth tables are used to determine the truth value of compound propositions based on the truth values of the individual propositions. Several examples are provided to illustrate how to construct truth tables for statements using various logical connectives. One example shows that p ∧ q is equal to q ∧ p through a truth table.
This document introduces basic concepts in propositional logic, including:
1. Propositions are declarative statements that are either true or false. Compound propositions consist of simple propositions connected by logical operators like AND and OR.
2. Truth tables define logical connectives like conjunction, disjunction, conditional, biconditional, and negation. Equivalences between statements can be shown through truth tables.
3. Logical implications can be proven without truth tables by showing that if the antecedent is true, the consequent must also be true. Dual statements and De Morgan's laws are also introduced.
This document provides an overview of international political economy (IPE). It discusses the meaning and nature of IPE, focusing on the political and economic dimensions. It outlines three major theoretical perspectives on IPE: mercantilism/economic nationalism, liberalism, and Marxism. Differences in national political economy systems are examined through case studies of the US, Japan, and Germany. Finally, core issues in IPE governance are discussed, specifically international trade and the role of the World Trade Organization.
The systematic study of science and religion was started in 1960s. Science and religion had been defacto Western science and Christianity for the past fifty years. One way to distinguish between science and religion is that Science concerns the natural world, whereas Religion concerns both the natural and the supernatural world. Barbour’s 4 models of science and religion interactions are Conflict, Independence, Dialogue, and Integration Models. Some philosophers suggest that Christianity was instrumental in catalyzing scientific revolution. Contemporary lack of scientific prominence is remarkable in the Islamic World. The two views of divine action are general divine action and special divine action. Evolutionary ethics & Implications of the cognitive science of religion are areas of increasing interest in science and religion.
This document provides an overview of an inclusive education course developed by several universities in Ethiopia. The course is intended to be required for all undergraduate students. It aims to teach students how to create inclusive environments and provide appropriate support to people with disabilities and vulnerabilities. The course objectives, chapter outlines, and learning outcomes are described. The first chapter focuses on defining key terms, describing various types and causes of disabilities, the history of inclusion, models of disability, and the impact of attitudes on inclusion.
This document discusses the electromagnetic spectrum and properties of light. It describes how light exhibits both wave-like and particle-like properties. The wave properties of light include frequency, wavelength, speed and amplitude. The particle properties include photons and the photoelectric effect. The document also covers the Bohr model of the hydrogen atom and how it led to the development of quantum theory, which explained atomic spectra and the dual wave-particle nature of matter and energy.
This document provides an overview of basic concepts in logic, including:
- Logic is defined as the science of reasoning, concerned with determining correct vs incorrect reasoning.
- Reasoning involves making inferences by drawing conclusions from premises. Arguments consist of statements where one is the conclusion and the others are premises intended to support the conclusion.
- Logic divides into deductive logic and inductive logic. Deductive logic examines necessarily truth-preserving arguments, while inductive logic examines arguments where the premises make the conclusion probable but not certain.
Ethiopian Law of Evidence Lecture Notes pptgetabelete
This document provides a summary of Berhe Gessesew's lecture on Chapter Two of Evidence Law regarding facts that may be proved other than by evidence. The summary is as follows:
1. Generally, parties have the burden of proving allegations, but some allegations do not require proof if admitted by the opposing party, presumed by the judge, or subject to judicial notice.
2. Admitted facts do not require proof because parties are assumed to know their own facts and courts aim to resolve disputes rather than prove undisputed facts. Admissions can be formal, informal, judicial or extra-judicial. More weight is given to formal admissions made to those with authority.
3. Presumptions shift or fix the
This document provides an introduction to logic. It outlines the objectives of studying logic as sharpening intellect, developing learning ability, strengthening understanding, and promoting clear thinking. Key benefits include supporting reasoning powers, distinguishing good from bad arguments, and learning principles of clear thinking. Logic is defined as the study of correct versus incorrect reasoning. Important concepts discussed include premises, propositions, arguments, sound versus unsound arguments, and laws of logic. The overall aim is to learn how to evaluate arguments.
CHAPTER ONE & TWO LOGIC AND PHILOSOPHY.pptxBarentuShemsu
This document provides an introduction to philosophy by outlining some of its key concepts and fields. It begins by defining philosophy as the love of wisdom and noting that philosophy deals primarily with fundamental issues rather than having a single subject matter. The document then outlines some of philosophy's major fields, including metaphysics, epistemology, axiology, and logic. For each field, it provides brief definitions and examples of the types of questions addressed. The document emphasizes that philosophy is an activity that encourages critical examination and reflection on life and reality.
This document discusses different research philosophies and approaches. It defines key characteristics of research paradigms such as ontology, epistemology, and axiology. It presents the "research onion" as a model of the layers within research philosophy. The main research philosophies discussed are positivism, realism, interpretivism, and pragmatism. Positivism is defined as trying to uncover objective truths about how the world works through logical deduction and empirical observation in order to predict and control outcomes.
Chapter One & Two.pptx introduction to emerging TechnologygadisaAdamu
This document provides an overview of emerging technologies and data science. It discusses key concepts like invention, innovation, and technology. It describes the four industrial revolutions and how they transformed society through innovations like the steam engine. Emerging technologies discussed include artificial intelligence, blockchain, and robotics. The role of data in powering emerging technologies is also examined. The document then shifts to discussing data science, defining data, information, and knowledge. It outlines the data processing cycle and common data types used in computer science.
This document provides an overview of philosophy and the philosophy of the human person. It defines philosophy as the systematic study of truth and the principles of beings through reason and faith. The document discusses the etymology and definitions of philosophy. It also outlines different levels and branches of wisdom and philosophy. In particular, it distinguishes between natural wisdom gained through reason and supernatural wisdom gained through faith. The document contrasts philosophy and theology, and discusses the importance and theoretical and practical branches of philosophy, including metaphysics, ethics, and aesthetics.
The document provides an overview of topics covered in the first week of a symbolic logic course, including:
1) The basic components of an argument such as premises, conclusions, and deductive validity.
2) Ways an argument can be weak, including having false premises or premises that don't support the conclusion.
3) Logical concepts like necessity, possibility, consistency, and logical equivalence.
4) The difference between formal languages like sentential logic and natural languages.
This document provides an overview of Western philosophy from ancient to medieval periods. It discusses some of the major philosophers from each era and their contributions. The ancient Greek philosophers like Socrates, Plato and Aristotle established the foundations of Western thought. Medieval philosophy integrated Greek rationalism with Christian theology. Major philosophers included Augustine of Hippo who wrote extensively on theology and philosophy. The document outlines some key characteristics of medieval philosophy like theocentrism, theodicy, and emphasis on God and faith.
Logic is the study of reasoning and correct thinking. It involves analyzing concepts, establishing general laws of
truth, and determining valid forms of argument. Logic is applicable to all fields as it provides standards for
consistent and evidence-based reasoning. It has wide scope and helps with social studies, engineering, mathematics,
science, and computer programming through modeling reality, simplifying complex problems, and representing
information processing in a logical way. Studying logic is important as it helps develop critical thinking skills
needed to make rational decisions, adapt to new situations, and form justifiable beliefs.
This document provides an overview of the course "Logic and Critical Thinking". It discusses the following key points:
1. The course covers 6 chapters, including introductions to logic, basic logic concepts, critical thinking, logical reasoning and fallacies, categorical propositions.
2. Chapter 1 defines philosophy as the love of wisdom and discusses its major fields including metaphysics, epistemology, axiology, and logic. It emphasizes that philosophy questions apparent truths.
3. Logic is the study of arguments and their structures. An argument consists of premises that provide support for a conclusion. Identifying premises and conclusions is important for evaluating arguments.
This document provides an introduction and overview of philosophy. It defines philosophy as the systematic study of fundamental human knowledge and the pursuit of wisdom. The document traces the origin of the term "philosophy" to ancient Greek roots meaning "love of wisdom". It outlines the main goals of philosophy as discovering the nature of truth and knowledge. The document also describes the scope and key branches of philosophy, and emphasizes philosophy's importance in clarifying beliefs, stimulating thinking, and developing analytical abilities.
The document discusses ancient Greek philosophers who lived before Socrates, known as the Pre-Socratics. Three philosophers from Miletus - Thales, Anaximander, and Anaximenes - believed the basic substance of the universe was water, infinity, and air respectively. Parmenides argued that change and motion are illusions and all is permanent. Heraclitus believed all is in flux. Empedocles said nature's source cannot be a single element. Anaxagoras believed seeds were ordered by intelligence. Pythagoras and his followers studied mathematics and believed the universe followed musical and numerical laws, discovering the Pythagorean theorem relating triangle side lengths.
Plato and the Pre-Socratic philosophers attempted to explain reality through reason alone, without reference to myth or religion. They debated questions about the fundamental nature of reality and whether it is one or many. Plato believed in eternal "Forms" that were the perfect essence of things in the world, which were imperfect copies. Aristotle rejected Plato's theory of Forms and argued reality is composed of both matter and form, with accidents being non-essential features and essences being defining characteristics. This document provides an overview of the major philosophers from Thales to Aristotle and some of the key debates around the nature of reality, knowledge, and ethics.
The document discusses the basics of logic including propositions, truth tables, and logical connectives. It defines a proposition as a statement that is either true or false. Compound propositions can be formed using logical connectives like AND, OR, XOR, NAND, and NOR. Truth tables are used to determine the truth value of compound propositions based on the truth values of the individual propositions. Several examples are provided to illustrate how to construct truth tables for statements using various logical connectives. One example shows that p ∧ q is equal to q ∧ p through a truth table.
This document introduces basic concepts in propositional logic, including:
1. Propositions are declarative statements that are either true or false. Compound propositions consist of simple propositions connected by logical operators like AND and OR.
2. Truth tables define logical connectives like conjunction, disjunction, conditional, biconditional, and negation. Equivalences between statements can be shown through truth tables.
3. Logical implications can be proven without truth tables by showing that if the antecedent is true, the consequent must also be true. Dual statements and De Morgan's laws are also introduced.
This document discusses logic and logical concepts such as propositions, truth tables, logical connectives, and logical equivalences. It defines key terms like proposition, truth value, conjunction, disjunction, negation, implication, biconditional, and provides examples of how to represent these concepts using truth tables. It also covers more complex topics like tautologies, compound statements, and the relationships between statements like converse, contrapositive and inverse. Exercises are provided to help understand applying these logical concepts.
This document discusses propositional logic and truth tables. It defines primitive and compound propositions. Logical connectives like negation, disjunction, and conjunction are explained. Propositional variables are used to represent statements that can be true or false. Truth tables list all possible combinations of true and false values for propositional variables and determine the truth value of compound statements formed from logical connectives. The number of rows in a truth table is determined by 2 to the power of the number of propositional variables. Several examples of truth tables are given for logical connectives like negation, disjunction, conjunction, implication, and biconditional.
Chapter 1 Logic of Compound Statementsguestd166eb5
This document introduces basic concepts in propositional logic and discrete mathematics including:
- Statements and their truth values
- Logical connectives such as negation, conjunction, disjunction, implication, biconditional
- Compound statements formed using logical connectives
- Truth tables to determine the truth values of compound statements
- Tautologies, contradictions and contingencies
- Negation, contrapositive, converse and inverse of conditional statements
- De Morgan's laws of negation for conjunction and disjunction
Examples are provided to illustrate key concepts and definitions throughout.
The document discusses propositional logic and truth tables. It defines statements as sentences that are either true or false. Examples of statements and non-statements are provided. The main logical connectives - and, or, if-then, if and only if, negation - are explained along with their symbols. Examples are given to illustrate how to determine the truth value of statements using truth tables for connectives involving two or more statements. The concepts of equivalent statements, tautologies, and using contradiction to check for tautologies are also explained with examples.
This document summarizes key concepts in logic:
1. Statements are declarative sentences that can be true or false. Logical connectives like "and", "or", and "not" are used to combine statements.
2. A tautology is a proposition that is always true, while a contradiction is always false. A contingency can be either true or false.
3. Logical equivalence means two statements always have the same truth value. Direct, converse, inverse, and contrapositive are types of logical implications between statements.
This document summarizes key concepts in logic:
1. Statements are declarative sentences that can be true or false. Logical connectives like "and", "or", and "not" are used to combine statements.
2. A tautology is a proposition that is always true, while a contradiction is always false. A contingency can be either true or false.
3. Logical equivalence means two statements always have the same truth value. Direct, converse, inverse, and contrapositive are types of logical implications between statements.
This document discusses truth tables and logical equivalences in propositional logic. It defines truth tables for various logical connectives like negation, conjunction, and disjunction. It also defines logical concepts like tautology, contradiction, and De Morgan's laws. Examples are provided to illustrate double negation, associative laws, distributive laws, and how to determine if a statement is a tautology or contradiction using truth tables.
This document provides an overview of propositional logic and logical operators. It defines basic concepts like propositions, logical connectives, and truth tables. Compound propositions are formed by combining one or more propositions using logical operators like conjunction, disjunction, negation, implication, equivalence, exclusive or, and others. Computer representations of logic using bits are also discussed, where true and false map to 1 and 0, and bitwise logic operators correspond directly to the logical connectives. Precedence rules for logical operators are defined.
The document discusses how truth tables can be used to determine the logical status of propositions and arguments. Truth tables assign truth values (True/False) to propositions based on the truth values of their component statements, allowing the logical status of single propositions and groups of propositions to be determined. The logical status can be tautology, contradiction, contingent, equivalent, satisfiable/consistent, or unsatisfiable/inconsistent depending on the truth values. Validity of arguments can also be determined from truth tables by checking if the conclusion is true in all rows where the premises are true. Examples of truth tables are provided to illustrate these concepts.
The document discusses truth tables and their use in determining the validity of arguments. It defines truth tables as listings of all possible combinations of true and false statements and the resulting truth values. It then explains conjunction, disjunction, negation, and logical equivalence through truth tables. Conjunction uses AND and is true only when both statements are true. Disjunction uses OR and is true if either statement is true. Negation reverses the truth value of a statement. Logical equivalence means statements have identical truth tables.
proposition, truth tables and tautology.pptxJayLagman3
This document defines key concepts in propositional logic including propositions, arguments, validity, and fallacies. It explains how to represent arguments using premises and conclusions. It also defines truth tables and how they are used to determine the truth value of compound statements for all possible combinations of true and false basic statements. Truth tables are provided for negation, conjunction, disjunction, implication, tautologies, and contradictions. An example shows how to identify a tautology and contradiction using truth tables. Exercises are provided to have the reader construct truth tables.
- The document discusses the mathematical foundations of computer science, including topics like mathematical logic, set theory, algebraic structures, and graph theory.
- It specifically focuses on mathematical logic, defining statements, atomic and compound statements, and various logical connectives like negation, conjunction, disjunction, implication, biconditional, and their truth tables.
- It also discusses logical concepts like tautologies, contradictions, contingencies, logical equivalence, and tautological implication through the use of truth tables and logical formulas.
This document discusses propositional logic and inference theory. It begins by defining propositions, truth values, and logical operators like conjunction, disjunction, negation, implication, biconditional, and their truth tables. It then discusses tautologies, contradictions, and logical equivalences. The document introduces rules of inference and methods for formal proof, including truth table technique and direct/indirect proofs. It provides examples of applying rules of inference and truth tables to evaluate arguments. The document outlines key concepts in propositional logic and inference theory.
The document provides an overview of propositional logic including:
1. It defines statements, logical connectives, and truth tables. Logical connectives like negation, conjunction, disjunction and others are explained.
2. It discusses various logical concepts like tautology, contradiction, contingency, logical equivalence, and logical implications.
3. It outlines propositional logic rules and properties including commutative, associative, distributive, De Morgan's laws, identity law, idempotent law, and transitive rule.
4. It provides an example of using truth tables to test the validity of an argument about bachelors dying young.
L4-IntroducClick to edit Master title styletion to logic.pptxFahmiOlayah
اسماعيل هنية: نحيClick to edit Master title styleي من ساندنا في #لبنان والعراق واليمن وكل من خرج في العالم أجمع نصرة لنا ولحقوقنا، ونحيي دولة جنوب إفريقيا على رفعها دعوى أمام محكمة العدل الدولية ضد جرائم الاحتلال بحق شعبنا
#غزه_تقاوم
The document discusses the semantics of propositional logic, including:
1) Defining logical formulas using a formal language and grammar;
2) Describing the meaning of logical connectives like conjunction and negation through truth tables;
3) Explaining how interpretations assign truth values to formulas based on the truth values of their components.
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Maths teachers guide For freshman course.pdf
1. 1
Unit ONE
Solutions/Answers to Exercises of page 6
1. Which of the following sentences are propositions? For those that are, indicate the truth value.
Solutions/Answers
a. 123 is a prime number. It is a proposition with truth value true T.
b. 0 is an even number. It is a proposition with truth value true T.
c. 𝑥2−4=0. It is not a proposition.
d. Multiply 5𝑥+2 by 3. It is not a proposition.
e. What an impossible question! It is not a proposition.
2. State the negation of each of the following statements.
Solutions/Answers
a. √2 is a rational number. √2 is not a rational number.
b. 0 is not a negative integer. 0 is a negative integer.
c. 111 is a prime number. 111 is not a prime number.
3. Let 𝑝: 15 is an odd number.
𝑞: 21 is a prime number.
Solutions/Answers
a. q
p : 15 is an odd number or 21 is a prime number. Truth value True
b. q
p : 15 is an odd number and 21 is a prime number. Truth value False
c. q
p
: 15 is not an odd number or 21 is a prime number. Truth value False
d. q
p
: 15 is an odd number and 21 is not a prime number. Truth value True
e. q
p : If 15 is an odd number then 21 is a prime number. Truth value False
f. p
q : If 21 is a prime number then 15 is an odd number. Truth value True
a. q
p
: If 15 is not an odd number then 21 is not a prime number. Truth value True
g. p
q
: If 21 is not a prime number then 15 is not an odd number. Truth value False
4. Complete the following truth table.
p q 𝒒 𝒑∧𝒒
T T F F
T F T T
F T F F
F F T F
2. 2
Solutions/Answers to Exercises of pages 11-12
1. For statements 𝑝, and 𝑟, use a truth table to show that each of the following pairs of statements is
logically equivalent.
a. (𝑝∧𝑞)⟺𝑝 and 𝑝⟹𝑞.
Therefore, (𝑝∧𝑞)⟺𝑝 and 𝑝⟹𝑞 are logically equivalent,
since the forth and the fifth columns have
the same truth values .
b. 𝑝⟹(𝑞∨𝑟) and 𝑞⟹(𝑝∨𝑟).
Therefore,
𝑝⟹(𝑞∨𝑟) and
𝑞⟹(𝑝∨𝑟) are
logically equivalent.
since the eighth and
the ninth columns have
the same truth values .
c. (𝑝∨𝑞)⟹𝑟 and (𝑝⟹𝑞)∧(𝑞⟹𝑟).
The truth values of the
combinations differ in
the third row of the
seventh and the eighth
columns where
(𝑝∨𝑞)⟹r is T and
(𝑝⟹𝑞)∧(𝑞⟹𝑟) is F so that
𝑝∨𝑞)⟹𝑟 and (𝑝⟹𝑞)∧(𝑞⟹𝑟) are
not logically equivalent.
d. 𝑝⟹(𝑞∨𝑟) and (𝑟)⟹(𝑝⟹𝑞).
The truth values of the combinations differ in the second and forth rows of the seventh and
the eighth columns so that 𝑝⟹(𝑞∨𝑟) and (𝑟)⟹(𝑝⟹𝑞) are not logically equivalent.
P Q 𝑝∧𝑞 (𝑝∧𝑞)⟺𝑝 𝑝⟹𝑞
T T T T T
T F F F F
F T F T T
F F F T T
P Q r 𝑞∨𝑟 P 𝑞 𝑝∨𝑟 𝑝⟹(𝑞∨𝑟) 𝑞⟹(𝑝∨𝑟)
T T T T F F T T T
T T F T F F F T T
T F T T F T T T T
T F F F F T F F F
F T T T T F T T T
F T F T T F T T T
F F T T T T T T T
F F F F T T T T T
P q r 𝑝∨𝑞 𝑝⟹𝑞 𝑞⟹𝑟 (𝑝∨𝑞)⟹𝑟 (𝑝⟹𝑞)∧(𝑞⟹𝑟)
T T T T T T T T
T T F T T F F F
T F T T F T T F
T F F T F T F F
F T T T T T T T
F T F T T F F F
F F T F T T T T
F F F F T T T T
P q r 𝑟 𝑞∨𝑟 𝑝⟹𝑞 𝑝⟹(𝑞∨𝑟) (𝑟)⟹(𝑝⟹𝑞)
T T T F T T T T
T T F T F T F T
T F T F T F T T
T F F T T F T F
F T T F T T T T
F T F T F T T T
F F T F T T T T
F F F T T T T T
3. 3
e. 𝑝⟹(𝑞∨𝑟) and ((𝑟)∧𝑝)⟹𝑞.
The truth values of the
combinations the propositions
of the seventh and
the eighth columns are
the same so that
𝑝⟹(𝑞∨𝑟) and (𝑟)⟹(𝑝⟹𝑞)
are logically equivalent.
2. For statements 𝑝, q, and 𝑟, show that the following compound statements are tautology.
a. 𝑝⟹(𝑝∨𝑞).
𝑝⟹(𝑝∨𝑞 is a tautology since all the possible combinations
in the last column are all T
b. (𝑝∧(𝑝⟹𝑞))⟹𝑞.
(𝑝∧(𝑝⟹𝑞))⟹𝑞 is a tautology since all
the possible combinations in the last
column are all T
c. ((𝑝⟹𝑞)∧(𝑞⟹𝑟))⟹(𝑝⟹𝑟).
((𝑝⟹𝑞)∧(𝑞⟹𝑟))⟹(𝑝⟹𝑟) is a tautology since all the possible combinations in the last
column are all T
p q r 𝑟 𝑞∨𝑟 (𝑟)∧𝑝 𝑝⟹(𝑞∨𝑟) ((𝑟)∧𝑝)⟹𝑞
T T T F T F T T
T T F T T T T T
T F T F T F T T
T F F T F T F F
F T T F T F T T
F T F T T F T T
F F T F T F T T
F F F T F F T T
p q 𝑝∨q 𝑝⟹(𝑝∨𝑞)
T T T T
T F T T
F T T T
F F F T
p q 𝑝⟹q 𝑝∧(𝑝⟹𝑞) (𝑝∧(𝑝⟹𝑞))⟹𝑞
T T T T T
T F F F T
F T T F T
F F T F T
p q r 𝑝⟹q 𝑞⟹𝑟 𝑝⟹𝑟 (𝑝⟹𝑞)∧(𝑞⟹𝑟) ((𝑝⟹𝑞)∧(𝑞⟹𝑟))⟹(𝑝⟹𝑟)
T T T T T T T T
T T F T F F F T
T F T F T T F T
T F F F T F F T
F T T T T T T T
F T F T F T F T
F F T T T T T T
F F F T T T T T
4. 4
(𝑝∧𝑞)∧(𝑝∧𝑞) is a contradiction since the last
column has all the truth values F
4. Write the contrapositive and the converse of the following conditional statements.
a. If it is cold, then the lake is frozen.
Contrapositive: If the lake is not frozen then it is not cold.
Converse: If the lake is frozen then it is cold.
b. If Solomon is healthy, then he is happy.
Contrapositive: If he is not happy then Solomon is not healthy.
Converse: If he is happy then Solomon is healthy.
c. If it rains, Tigist does not take a walk.
Contrapositive: If Tigist takes a walk, it doesn't rain.
Converse: If Tigist doesn't take a walk, it rains.
5. Let 𝑝 and 𝑞 be statements. Which of the following implies that 𝑝∨𝑞 is false?
𝑝∨𝑞 is false means p is True and q is False so that:
a. 𝑝∨𝑞 is false.
𝑝 is F and 𝑞 is T which implies F ∨ T which implies T so that 𝑝∨𝑞 is T
b. 𝑝∨𝑞 is true.
This means F or F which is F
c. 𝑝∧𝑞 is true.
This means F ∧ T which is F
d. 𝑝⟹𝑞 is true.
This means T implies F which is F
e. 𝑝∧𝑞 is false.
This means T and F which is F
So for in general for question number 5 , the answers are b, c, d, e
3. For statements 𝑝 and 𝑞, show that (𝑝∧𝑞)∧(𝑝∧𝑞)
is a contradiction.
P q 𝑞 𝑝∧q 𝑝∧𝑞 (𝑝∧𝑞)∧(𝑝∧𝑞)
T T F F T F
T F T T F F
F T F F F F
F F T F F F
5. 5
6. Suppose that the statements 𝑝, q, 𝑟, and 𝑠 are assigned the truth values 𝑇,,, and 𝑇, respectively.
Find the truth value of each of the following statements.
a. (𝑝∨𝑞)∨𝑟.
(T∨F)∨F which gives T∨F and gives T
b. 𝑝∨(𝑞∨𝑟).
T∨(F∨F) which gives T∨F and gives T
c. 𝑟⟹(𝑠∧𝑝)
F⟹(T∧T) which gives F⟹T which also gives T
d. 𝑝⟹(𝑟⟹𝑠).
T⟹(F⟹T) which gives T⟹T which also gives T
e. 𝑝⟹(𝑟∨𝑠).
T⟹(F∨T) which gives T⟹T which lso gives T
f. (𝑝∨𝑟)⟺(𝑟∧Ø𝑠).
(T∨F)⟺(F∧F) which gives T⟺F which also gives F
g. (𝑠⟺𝑝)⟹(Ø𝑝∨𝑠).
(T⟺T)⟹(F∨T) which gives T⟺T which also gives T
h. (𝑞∧Ø𝑠)⟹(𝑝⟺𝑠).
(F∧F)⟹(T⟺T) which gives F⟹T which also gives T
i. (𝑟∧𝑠)⟹(𝑝⟹(Ø𝑞∨𝑠)).
(F∧T)⟹(T⟹(T∨T)) which gives F⟹(T⟹T) which also gives T⟹T and also gives T
j. (𝑝∨Ø𝑞)∨𝑟⟹(𝑠∧Ø𝑠).
(T∨T)∨F⟹(𝑠∧Ø𝑠) which gives F⟹(T⟹T) which also gives T⟹T and also gives T
7. Suppose the value of 𝑝⟹𝑞 is 𝑇; what can be said about the value of 𝑝∧𝑞⟺𝑝∨𝑞?
𝑝⟹𝑞 is 𝑇 has three cases which are p is T and q is T; p is F and q is T; ; p is F and q is F
Case 1):- When p is T and q is T then 𝑝∧𝑞⟺𝑝∨𝑞 means F∧T⟺T∨T which is F ⟺ T which is F
Case 2):- When p is F and q is T then 𝑝∧𝑞⟺𝑝∨𝑞 means T∧T⟺T∨T which is T ⟺ T which is T
Case 3):- When p is F and q is F then 𝑝∧𝑞⟺𝑝∨𝑞 means T∧F⟺F∨F which is F ⟺ T which is F
Therefore, 𝑝∧𝑞⟺𝑝∨𝑞 has a truth value of either T or F
6. 6
8. a. Suppose the value of 𝑝⟺𝑞 is 𝑇; what can be said about the values of 𝑝⟺𝑞 and 𝑝⟺𝑞?
𝑝⟺𝑞 is 𝑇 means p and q have the same truth value
Which means p is T and q is T or p is F and q is F
Therefore, when p is T and q is T then 𝑝⟺𝑞 means T⟺F and 𝑝⟺𝑞 means F⟺T
which means F
And, when p is F and q is F then 𝑝⟺𝑞 means T⟺T and 𝑝⟺𝑞 T⟺T which means T
b. Suppose the value of 𝑝⟺𝑞 is 𝐹; what can be said about the values of 𝑝⟺𝑞 and 𝑝⟺𝑞?
𝑝⟺𝑞 is 𝐹 means p and q have different truth values
which means p is T and q is F or p is F and q is T
Therefore, when p is T and q is F then 𝑝⟺𝑞 means T⟺T and 𝑝⟺𝑞 means F⟺F
which means T
9. Construct the truth table for each of the following statements.
a. 𝑝⟹(𝑝⟹𝑞).
P q 𝑝⟹𝑞 𝑝⟹(𝑝⟹𝑞)
T T T T
T F F F
F T T T
F F T T
b. (𝑝∨𝑞)⟺(𝑞∨𝑝).
p q 𝑝∨𝑞 𝑞∨𝑝 (𝑝∨𝑞)⟺(𝑞∨𝑝)
T T T T T
T F T T T
F T T T T
F F F F T
c. 𝑝⟹(𝑞∧𝑟).
p q r 𝑞∧𝑟 (𝑞∧𝑟) 𝑝⟹(𝑞∧𝑟)
T T T T F F
T T F F T T
T F T F T T
T F F F T T
F T T T F T
F T F F T T
F F T F T T
F F F F T T
7. 7
d. (𝑝⟹𝑞)⟺( 𝑝∨𝑞).
p q 𝑝 𝑝⟹𝑞 𝑝∨𝑞 (𝑝⟹𝑞)⟺( 𝑝∨𝑞)
T T F T T T
T F F F F T
F T T T T T
F F T T T T
e. (p⟹(q∧r))∨(p∧q).
p q r p q∧r p∧q p⟹(q∧r) (p⟹(q∧r))∨( p∧q)
T T T F T F T T
T T F F F F F F
T F T F F F F F
T F F F F F F F
F T T T T T T T
F T F T F T T T
F F T T F F T T
F F F T F F T T
f. (𝑝∧𝑞)⟹((𝑞∧𝑞)⟹(𝑟∧𝑞)).
p Q r q 𝑝∧𝑞 𝑞∧𝑞 𝑟∧𝑞 ((𝑞∧𝑞)⟹(𝑟∧𝑞)) (𝑝∧𝑞)⟹((𝑞∧𝑞)⟹(𝑟∧𝑞))
T T T F T F T T T
T T F F T F F T T
T F T T F F T T T
T F F T F F F T T
F T T F T F T T T
F T F F T F F T T
F F T T F F T T T
F F F T F F F T T
10. For each of the following determine whether the information given is sufficient to
decide the truth value of the statement. If the information is enough, state the truth
value. If it is insufficient, show that both truth values are possible.
a. (𝑝⟹𝑞)⟹𝑟, where 𝑟=𝑇.
r =T is sufficient to determine the truth value of (𝑝⟹𝑞)⟹𝑟
since when r =T, whatever the truth value of 𝑝⟹𝑞 is, r =T gives (𝑝⟹𝑞)⟹𝑟 is T
8. 8
b. 𝑝∧(𝑞⟹𝑟), where 𝑞⟹𝑟=𝑇.
𝑞⟹𝑟=𝑇 is insufficient to determine the truth value of 𝑝∧(𝑞⟹𝑟)
Since when 𝑞⟹𝑟=𝑇, the truth value of is 𝑝∧(𝑞⟹𝑟) either T or F depending the
truth value of P, i.e., when 𝑞⟹𝑟=𝑇; p is T, then 𝑝∧(𝑞⟹𝑟) is T
when 𝑞⟹𝑟=𝑇; p is F, then 𝑝∧(𝑞⟹𝑟) is F
c. 𝑝∨(𝑞⟹𝑟), where 𝑞⟹𝑟=𝑇.
𝑞⟹𝑟=𝑇 is sufficient to determine the truth value of 𝑝∨(𝑞⟹𝑟)
since when 𝑞⟹𝑟=𝑇, whatever the truth value of 𝑝 is, ⟹𝑟=𝑇 gives 𝑝∨(𝑞⟹𝑟) is T
d. (𝑝∨𝑞)⟺(𝑝∧𝑞), where 𝑝∨𝑞=𝑇.
𝑝∨𝑞=𝑇 is sufficient to determine the truth value of (𝑝∨𝑞)⟺(𝑝∧𝑞)
Since 𝑝∨𝑞=𝑇 means (𝑝∨𝑞)is F and (𝑝∧𝑞) (𝑝∨𝑞) is F
so that (𝑝∧𝑞) (𝑝∨𝑞) is T
e. (𝑝⟹𝑞)⟹(𝑞⟹𝑝), where 𝑞=𝑇.
𝑞=𝑇 is sufficient to determine the truth value of (𝑝⟹𝑞)⟹(𝑞⟹𝑝)
Since 𝑞=𝑇 means 𝑝⟹𝑞 is T ; 𝑞 is F and 𝑞⟹𝑝 is T
and hence (𝑝⟹𝑞)⟹(𝑞⟹𝑝) is T
f. (𝑝∧𝑞)⟹(𝑝∨𝑠), where 𝑝=𝑇 and 𝑠=𝐹.
𝑝=𝑇 and 𝑠=𝐹 are sufficient to determine the truth value of (𝑝∧𝑞)⟹(𝑝∨𝑠)
Since 𝑝=𝑇 and 𝑠=𝐹 means 𝑝∨𝑠 is T and (𝑝∧𝑞)⟹(𝑝∨𝑠) is T whatever the truth
value of 𝑝∧𝑞 is.
Even, only 𝑝=𝑇 is sufficient to determine the truth value of (𝑝∧𝑞)⟹(𝑝∨𝑠),
Since 𝑝=𝑇 means 𝑝∨𝑠 is T for any truth value of s,
and 𝑝∨𝑠 is T means (𝑝∧𝑞)⟹(𝑝∨𝑠) is T for any truth value of 𝑝∧𝑞
9. 9
Solutions/Answers to Exercises of pages 19-20
1. In each of the following, two open statements 𝑃(𝑥,𝑦) and 𝑄(𝑥,𝑦) are given, where the domain
of both 𝑥 and 𝑦 is . Determine the truth value of (𝑥,)⟹𝑄(𝑥,𝑦) for the given values of 𝑥 and 𝑦.
a. (𝑥,):𝑥2−𝑦2=0. and 𝑄(𝑥,𝑦):𝑥=𝑦. (𝑥,𝑦)∈*(1,−1),(3,4),(5,5)+.
(𝑥,):𝑥2−𝑦2=0 is F since when (𝑥,)=(3,4); 𝑥2−𝑦2= 32−42=9-16=-7≠0 and
(𝑥,):𝑥=𝑦 is F since when (𝑥,)=(3,4); 3=4 is F
Therefore, (𝑥,𝑦)⟹𝑄(𝑥,𝑦) means F⟹F which is T
b. 𝑃(𝑥,𝑦):|𝑥|=|𝑦|. and 𝑄(𝑥,𝑦):𝑥=𝑦. (𝑥,𝑦)∈*(1,2),(2,−2),(6,6)+.
(𝑥,):|𝑥|=|𝑦| is F since when (𝑥,)=(1,2);|1|≠|2| and
(𝑥,):𝑥=𝑦 is F since when (𝑥,)=(1,2);1≠2
Therefore, (𝑥,𝑦)⟹𝑄(𝑥,𝑦) means F⟹F which is T
c. (𝑥,):𝑥2+𝑦2=1. and 𝑄(𝑥,𝑦):𝑥+𝑦=1. (𝑥,𝑦)∈*(1,−1), (−3,4), (0,−1), (1,0)}.
(𝑥,): 𝑥2+𝑦2=1 is F since when (𝑥,)= (−3,4); (-3)2+42=1 is F and
Q(𝑥,𝑦): 𝑥+𝑦=1 is F since when (𝑥,𝑦)= (1,0);0=1 is F
Therefore, (𝑥,𝑦)⟹𝑄(𝑥,𝑦) means F⟹F which is T
2. Let 𝑂 denote the set of odd integers and let (𝑥):𝑥2+1 is even, and (𝑥):𝑥2 is even. be open statements over the
domain 𝑂. State (∀𝑥∈𝑂)(𝑥) and (∃𝑦∈𝑂)𝑄(𝑥) in words.
For every odd integer x , 𝑥2+1 is even
and there is a an odd integer y such that y2 is even
3. State the negation of the following quantified statements.
a. For every rational number 𝑟, the number 1/𝑟 is rational.
Not for every rational number , the number 1/𝑟 is rational.
OR, For some rational number , the number 1/𝑟 is not rational.
b. There exists a rational number 𝑟 such that 𝑟2=2.
Not there exists a rational number 𝑟 such that 𝑟2=2.
OR, For every a rational number , 𝑟2≠2.
10. 10
4. Let (𝑛):
3
6
-
5n
is an integer. be an open sentence over the domain . Determine, with explanations, whether the
following statements are true or false:
a. (∀𝑛∈ℤ)(𝑛).
(∀𝑛∈ℤ)(𝑛) is False since for
3
1
-
3
6
-
5x1
3
6
-
5n
1
that
such
Z
n is not an integer.
b. (∃𝑛∈ℤ)(𝑛).
(∃𝑛∈ℤ)(𝑛) is True since 2
3
6
-
3
6
-
5x0
3
6
-
5n
0
that
such
Z
n is an integer.
5. Determine the truth value of the following statements.
a. (∃𝑥∈ℝ)(𝑥2
−𝑥=0).
True (when we take x=1 ; 𝑥2
−𝑥=0 which means 12
−1=1−1=0
b. (∀𝑥∈ℕ)(𝑥+1≥2).
False (when we take x=−10 ; 𝑥+1=−10+1= − 9≥ 2 is false)
a. (∀𝑥∈ℝ)( 2
x =𝑥).
False (when we take x=−1 ; 1
1
1
1
2
2
x
x )
b. (∃𝑥∈ℚ)(3𝑥2
−27=0).
True (when we take x=3 ; 3𝑥2
−27=3(3 2
)−27=27−27=0)
c. (∃𝑥∈ℝ)(∃𝑦∈ℝ)(𝑥+𝑦+3=8).
True (when we take x=3 & y=2 ; 𝑥+𝑦+3=3+2+3=8)
d. (∃𝑥∈ℝ)(∃𝑦∈ℝ)(𝑥2
+𝑦2
=9).
True (when we take x= 8 & y=1 ; 𝑥2
+𝑦2
= 2
8 + 12
= 8+1= 9)
e. (∀𝑥∈ℝ)(∃𝑦∈ℝ)(𝑥+𝑦=5).
True (since x is chosen first, we can determine a single y=5-x so that x+y = x+5-x=5)
f. (∃𝑥∈ℝ)(∀𝑦∈ℝ)(𝑥+𝑦=5)
True (since y is chosen first, and then we vary x arbitrarily after y is determined, 𝑥+𝑦=5 can’t be
true like when we fix y=0, x could be any real number like 100, 0, -21 that makes 𝑥+𝑦=5 false )
6. Consider the quantified statement
For every 𝑥∈𝐴 and 𝑦∈𝐴, 𝑥𝑦−2 is prime.
where the domain of the variables 𝑥 and 𝑦 is 𝐴={3,5,11}.
a. Express this quantified statement in symbols.
Let P(x, y): 𝑥𝑦−2 is prime
(∀𝑥∈A) (∀𝑦∈A)P(x, y) or (∀𝑥∈A) (∀𝑦∈A) ( 𝑥𝑦−2 is prime).
b. Is the quantified statement in (a) true or false? Explain.
True
11. 11
c. [(∀𝑥∈A) (∀𝑦∈A)P(x, y)] or [(∀𝑥∈A) (∀𝑦∈A) ( 𝑥𝑦−2 is prime].
OR [(𝑥∈A) (𝑦∈A) P(x, y)] or [(𝑥∈A) (𝑦∈A) ( 𝑥𝑦−2 is prime].
d. Is the negation of the quantified in (a) true or false? Explain.
It is False
since (∀𝑥∈A) (∀𝑦∈A)P(x, y) or (∀𝑥∈A) (∀𝑦∈A) ( 𝑥𝑦−2 is prime). is True
and the negation of True is False
7. Consider the open statement (𝑥,):𝑥/𝑦<1. where the domain of 𝑥 is 𝐴={2,3,5} and the
domain of 𝑦 is 𝐵 = {2,4,6}.
a. State the quantified statement (∀𝑥 ∈ 𝐴)(∃𝑦 ∈ 𝐵)𝑃(𝑥, 𝑦) in words.
For every 𝑥∈𝐴 and for some 𝑦∈B, 𝑥/𝑦<1.
b. Show quantified statement in (a) is true.
Since y is determined just after x is chosen like when x=5, we can determine y=6 so that
𝑥/𝑦= 5/6<1 is true.
8. Consider the open statement (𝑥, 𝑦): 𝑥 − 𝑦 < 0. where the domain of 𝑥 is 𝐴 = {3,5,8} and the domain
of 𝑦 is 𝐵 = {3,6,10}.
a. State the quantified statement (∃𝑦 ∈ 𝐵)(∀𝑥 ∈ 𝐴)𝑃(𝑥, 𝑦) in words.
There is a 𝑦 in 𝐵 which stands to every x in A such that 𝑥 − 𝑦 < 0
b. Show quantified statement in (a) is true.
Let y=10, then for each x in A,
i.e., x=3, x=5, or x=8 , x−y = 8−10 = −2< 0, x−y =5−10=−5< 0, , x−y =3−10=−7< 0
12. 12
Solutions/Answers to Exercises of pages 24-25
1. Use the truth table method to show that the following argument forms are valid.
i. 𝑝⟹𝑞, 𝑞 ├ 𝑝.
It is to check whether ((𝑝⟹𝑞)𝑞) 𝑝 is tautology or not
p q 𝑝 𝑞 𝑝⟹𝑞 (𝑝⟹𝑞)𝑞 ((𝑝⟹𝑞)𝑞) 𝑝
T T F F T T T
T F F T T F T
F T T F F F T
F F T T T F T
Since ((𝑝⟹𝑞)𝑞) 𝑝 is a tautology, 𝑝⟹𝑞,𝑞 ├ 𝑝 is valid
ii. 𝑝⟹𝑝,,⟹𝑞 ├ 𝑟.
It is to check whether {[(𝑝⟹𝑝)𝑝](𝑟⟹𝑞)} 𝑟 is tautology or not
p q r p 𝑝⟹𝑝 (𝑝⟹𝑝)𝑝 𝑟⟹𝑞 [(𝑝⟹𝑝)𝑝](𝑟⟹𝑞) {[(𝑝⟹𝑝)𝑝](𝑟⟹𝑞)} 𝑟
T T T F F F T F T
T T F F F F T F T
T F T F F F F F T
T F F F F F T F T
F T T T T F T F T
F T F T T F T F T
F F T T T F F F T
F F F T T F T F T
Since {[(𝑝⟹𝑝)𝑝](𝑟⟹𝑞)} 𝑟 is a tautology, 𝑝⟹𝑝,𝑝,𝑟⟹𝑞 ├ 𝑟 is valid
iii. 𝑝⟹𝑞, 𝑟⟹𝑞 ├ 𝑟⟹𝑝.
It is to check whether {[(𝑝⟹𝑞) (𝑟⟹𝑞)]}⟹(𝑟⟹𝑝) is tautology or not
p q r p q r 𝑝⟹q r⟹q 𝑟⟹p (𝑝⟹𝑞) (𝑟⟹𝑞) {[(𝑝⟹𝑞) (𝑟⟹𝑞)]}⟹(𝑟⟹𝑝)
T T T F F F T T T T T
T T F F F T T F F F T
T F T F T F F T T F T
T F F F T T F T F F T
F T T T F F T T T T T
F T F T F T T F T F T
F F T T T F T T T T T
F F F T T T T T T T T
Since {[(𝑝⟹𝑞) (𝑟⟹𝑞)]}⟹(𝑟⟹𝑝)is a tautology, 𝑝⟹𝑞, 𝑟⟹𝑞 ├ 𝑟⟹𝑝 is valid
iv. 𝑟∨𝑠,( 𝑠⟹𝑝)⟹𝑟 ├ 𝑝.
It is to check whether{ (𝑟∨𝑠)[( 𝑠⟹𝑝)⟹𝑟]} ⟹𝑝 is tautology or not
p r s p r s 𝑟∨𝑠 𝑠⟹𝑝 ( 𝑠⟹𝑝)⟹𝑟 (𝑟∨𝑠)[( 𝑠⟹𝑝)⟹𝑟] { (𝑟∨𝑠)[( 𝑠⟹𝑝)⟹𝑟]} ⟹𝑝
T T T F F F F T T F T
T T F F F T T T T T T
T F T F T F T T F F T
T F F F T T T T F F T
F T T T F F F T T F T
F T F T F T T F T T T
F F T T T F T T F F T
F F F T T T T F T T T
Since { (𝑟∨𝑠)[( 𝑠⟹𝑝)⟹𝑟]} ⟹𝑝 is a tautology, 𝑟∨𝑠,( 𝑠⟹𝑝)⟹𝑟 ├ 𝑝 is valid
13. 13
v. 𝑝⟹𝑞, 𝑝⟹𝑟,⟹𝑠├ 𝑞⟹𝑠.
It is to check whether [(𝑝⟹𝑞) (𝑝⟹𝑟) (𝑟⟹𝑠)] ⟹(𝑞⟹𝑠) is tautology or not
p Q r s 𝑝⟹𝑞 𝑝⟹𝑟 𝑟⟹s (𝑝⟹𝑞) (𝑝⟹𝑟) (𝑟⟹𝑠) 𝑞⟹𝑠 [(𝑝⟹𝑞) (𝑝⟹𝑟) (𝑟⟹𝑠)] ⟹(𝑞⟹𝑠)
T T T T T T T T T T
T T T F T T F F T T
T T F T T T T T T T
T T F F T T T T T T
T F T T F T T F T T
T F T F F T F F F T
T F F T F T T F T T
T F F F F T T F F T
F T T T T T T T T T
F T T F T T F F T T
F T F T T F T F T T
F T F F T F T F T T
F F T T T T T T T T
F F T F T T F F F T
F F F T T F T F F T
F F F F T F T F F T
Since [(𝑝⟹𝑞) (𝑝⟹𝑟) (𝑟⟹𝑠)] ⟹(𝑞⟹𝑠) is a tautology, 𝑝⟹𝑞, 𝑝⟹𝑟,𝑟⟹𝑠├ 𝑞⟹𝑠 is valid
2. For the following argument given a, b and c below:
i. Identify the premises.
ii. Write argument forms.
iii. . Check the validity.
Answers
i. Identify the premises.
a. The following are premises:
If he studies medicine, he will get a good job.
If he gets a good job, he will get a good wage.
He did not get a good wage.
b. The following are premises:
If the team is late, then it cannot play the game.
If the referee is here, then the team is can play the game.
The team is late.
c. The following are premises:
If the professor offers chocolate for an answer, you answer the professor’s question.
The professor offers chocolate for an answer.
ii. Write argument forms.
a) Let p: he studies medicine
q: he will get a good job
r: he will get a good wage
Then
𝑝⟹q
q⟹r
r
p
14. 14
b) Let p: the team is late
q: the team can play the game
r: the referee is here
C) Let p: the professor offers chocolate for an answer
q: you answer the professor’s question
iii. Check the validity.
a)
Since [(𝑝⟹q)(q⟹r)r]p is not a tautology, the argument formed is not valid.
b)
Since [(𝑝⟹q) q]⟹r is not a tautology, the argument formed is not valid.
C)
Since [(𝑝⟹q) q]⟹q is a tautology, the argument formed is valid.
p q r p r 𝑝⟹q q⟹r (𝑝⟹q)(q⟹r)r [(𝑝⟹q)(q⟹r)r]p
T T T F F T T F T
T T F F T T F F T
T F T F F F T F T
T F F F T F T F T
F T T T F T T F T
F T F T T T F F T
F F T T F T T F T
F F F T T T T T F
p q r q r 𝑝⟹q (𝑝⟹q)p [(𝑝⟹q) q]⟹r
T T T F F F F T
T T F F T F F T
T F T T F T T F
T F F T T T T T
F T T F F F F T
F T F F T F F T
F F T T F F F T
F F F T T F F T
p q 𝑝⟹q (𝑝⟹q)p [(𝑝⟹q) q]⟹q
T T T T T
T F F F T
F T T F T
F F T F T
𝑝⟹q
q⟹r
r
p
𝑝⟹q
p
r
𝑝⟹q
p
q
Then
𝑝⟹q
p
r
Then
𝑝⟹q
p
q
15. 15
3. Give formal proof to show that the following argument forms are valid.
a. 𝑝⟹𝑞, 𝑞 ├ 𝑝.
We want to show the following is valid:
p
q
p
q
(1) q is true premise
(2) 𝑝⟹𝑞 is true premise
(3) q⟹p is true Contrapositive of (2)
(4) Therefore,
p
p
q
q
is valid by Modes Ponens and hence
p
q
p
q
is valid
b. 𝑝⟹𝑞,𝑝,𝑟⟹𝑞 ├ 𝑟.
We want to show the following is valid:
r
q
r
q
p
p
(1) 𝑝⟹𝑞 is true premise
(2) 𝑝 is true premise
(3)
q
q
p
p
is valid Modes Ponens (Meaning q
is true)
(4)
r
q
r
q
r
q
r
q
p
p
from (3) above
(5)
r
q
r
q
is valid by Modes Tollens and hence
r
q
r
q
p
p
is valid
16. 16
c. 𝑝⟹𝑞,𝑟⟹𝑞 ├ 𝑟⟹𝑝.
We want to show the following is valid:
p
r
q
p
q
r
(1) q
r
is true Premise
(2) q
p is true Premise
(3) p
q
is true Contrapositive of (2) above
(4) Therefore, to check the validity of:
p
r
q
p
q
r
is the same as checking the validity of
p
r
p
q
q
r
(5) So,
p
r
p
q
q
r
is valid by principle of Syllogism
d. 𝑟∧𝑠,(𝑠⟹𝑝)⟹𝑟 ├ 𝑝.
We want to show the following is valid:
p
r
p
s
s
r
(1) 𝑟∧𝑠 is true Premise
(2) 𝑟 is true Principle of detachment from (1) above
(3) So,
p
r
p
s
s
r
becomes
p
r
p
s
r
from (2) above
(4)
p
s
r
p
s
r
by Modes Tollens
(5)
p
s
p
s
p
s
by equivalence property
(6)
p
is true Principle of detachment from (5) above
(7) Therefore,
p
r
p
s
s
r
is valid
17. 17
e. 𝑝⟹,𝑝⟹𝑟,𝑟⟹𝑠 ├ 𝑞⟹𝑠.
We want to show the following is valid:
s
q
s
r
r
p
p
which is wrong (incomplete) question
f. 𝑝∨𝑞,𝑟⟹𝑝,𝑟 ├ 𝑞.
We want to show the following is valid:
q
r
p
r
q
p
(1) r is true premise
(2) p
r is true premise
(3)
p
p
r
r
is valid Modes Ponens
(4) q
p
q
p By principle of equivalence
(5)
q
r
p
r
q
p
becomes
q
q
p
p
(6)
q
q
p
p
is valid by Modes Ponens
(7) 𝑝∧𝑞,(𝑞∨𝑟)⟹ 𝑝 ├ 𝑟.
(8) 𝑝⟹(𝑞∨𝑟),𝑟,𝑝 ├ 𝑞.
18. 18
i. 𝑞⟹𝑝,⟹𝑝,𝑞 ├ 𝑟.
We want to show the following is valid:
r
q
p
r
p
q
(1) q
is true Premise
(2) p
q
is true Premise
(3)
p
p
q
q
is valid Modes Ponens
(4)
r
q
p
r
p
q
becomes
r
p
r
p
(5)
r
p
r
p
is valid by Modes Tollens
4. Prove the following are valid arguments by giving formal proof.
a. If the rain does not come, the crops are ruined and the people will starve. The crops are not ruined or the
people will not starve. Therefore, the rain comes.
Let p: The rain comes
q: The crops are ruined
r: The people will starve
Hence:
p
r
q
r)
(q
p
(1) r
q
is true premise
(2) r)
(q
is true properties of equivalence
19. 19
(3)
p
r
q
r)
(q
p
becomes
p
r
q
r)
(q
p
or
p
r)
(q
p
r
q
(4)
p
r)
(q
p
r
q
is valid Modes Tollens
c. If the team is late, then it cannot play the game. If the referee is here then the team can play the game.
The team is late. Therefore, the referee is not here.
Let p: The team is late
q: It can play the game
r: the referee is here
Hence:
r
p
q
r
q
p
(1) p is true Premise
(2) q
p
is true Premise
(3)
q
q
p
p
is valid Modes Ponens
(4)
r
p
q
r
q
p
becomes
r
q
q
r
or
r
q
r
q
(5)
r
q
r
q
is valid Modes Tollens
20. 20
Solutions/Answers to Exercises of pages 29-31
1. Which of the following are sets?
a. 1,2,3
Not a set since the elements are not contained by braces.
b. {1,2},3
Not a set since the element 3 is not contained by braces.
d. {{1},2},3
Not a set since the element 3 is not contained by braces.
e. {1,{2},3}
It is a set which has three elements 1,{2},3.
f. {1,2,a,b}.
It is a set which has four elements 1,2,a, b.
Generally, the objects in 1a, 1b, 1c are not sets but 1d and 1e are sets.
2. Which of the following sets can be described in complete listing, partial listing and/or set-builder methods?
Describe each set by at least one of the three methods.
The sets can be described as followed by each question:
a. The set of the first 10 letters in the English alphabet.
(i) Complete listing method as: {a, b, c, d, e, f, g, h, i, j}
b. The set of all countries in the world.
(i) Set-builder method: {x: x is a country in the world}
c. The set of students of Addis Ababa University in the 2018/2019 academic year.
(i) Set-builder method:
{x: x is a student in Addis Ababa University in the 2018/2019 academic year}
d. The set of positive multiples of 5.
(i) Set-builder method: {x: x is a positive multiple of 5}
(ii) partial listing method: {5, 10, 15, …}
e. The set of all horses with six legs.
(i) Set-builder method: {x: x is a horse with six legs}
3. Write each of the following sets by listing its elements within braces.
a. 𝐴={𝑥∈ℤ:−4<𝑥≤4+ 𝐴=*−3, −2, −1, 0, 1, 2, 3, 4 }
b. 𝐵={𝑥∈ℤ:𝑥2<5} B={0, 1, 4 }
c. 𝐶={𝑥∈ℕ:𝑥3<5} C={1}
21. 21
d. ={𝑥∈ℝ:𝑥2−𝑥=0} D={0, 1}
e. ={𝑥∈ℝ:𝑥2+1=0}. E={ }
4. Let 𝐴 be the set of positive even integers less than 15. Find the truth value of each of the following.
a. 15∈𝐴 False
b. −16∈𝐴 False
c. 𝜙∈𝐴 True
d. 12⊂𝐴 False
e. {2,8,14}∈𝐴 False
f. *2,3,4+⊆𝐴 False
g. {2,4}∈𝐴 False
h. 𝜙⊂𝐴 True
i. {246}⊆𝐴 False
5. Find the truth value of each of the following and justify your conclusion.
a. 𝜙⊆𝜙 True since empty set is the subset of any set.
b. *1,2+⊆*1,2+ True since any set is the subset of itself
c. 𝜙∈𝐴 for any set A False since if A={1, 2} the elements of A are only 1 and 2 only but not 𝜙
d. {𝜙}⊆𝐴, for any set A
False since if A={1} then subsets of A are written as: 𝜙 ⊆𝐴 and {1}⊆𝐴 only but not 𝜙}⊆𝐴
e. 5, 7⊆{5, 6, 7, 8} False since 5, 7∈{5, 6, 7, 8}but not 5, 7⊆{5, 6, 7, 8}
f. 𝜙∈{{𝜙}} False since 𝜙∈{𝜙}but not 𝜙∈{{𝜙}}
g. For any set 𝐴, 𝐴⊂𝐴
False since for any two sets A & B, B ⊂A means (x)(x∈B x∈A and y ∈A but y B)
Or B ⊂A B ⊆A but A≠ B
h. {𝜙} =𝜙 False since {𝜙} has one element which is 𝜙 but 𝜙 has no any element
6. For each of the following set, find its power set.
The proper subsets are described below each question:
a. {𝑎𝑏} 𝜙
b. {1, 1.5} 𝜙, {1}, {1.5}
c. {𝑎, 𝑏} 𝜙, {𝑎}, {𝑏}
d. {𝑎, 0.5, 𝑥} 𝜙, {𝑎}, {0.5}, { }
7. How many subsets and proper subsets do the sets that contain exactly 1,2,3,4,8,10 and 20 elements have?
To determine the number of subsets and proper subsets of the given set, first it is better to determine the
number of elements of the set which is 7.
Therefore the number of subsets is found by 27
= 128
and the number of proper subsets is found by (27
) – 1 = 128 – 1 = 127
22. 22
8. If 𝑛 is a whole number, use your observation in Problems 6 and 7 to discover a formula for the number of
subsets of a set with 𝑛 elements. How many of these are proper subsets of the set?
Number of subsets a set with 𝑛 elements is 2n
Number of subsets a set with 𝑛 elements is (2n
) – 1
9. Is there a set A with exactly the following indicated property?
a. Only one subset
Yes, A=𝜙 has exactly one subset which is 𝜙 itself,
or n(𝜙) = n(A) =0 and number of subsets of A is 20=1
b. Only one proper subset
Yes, A={1} has exactly one proper subset which is 𝜙,
or n(A)= 1 and number of proper subsets of A = (21) – 1 = 2 – 1 = 1
c. Exactly 3 proper subsets
Yes, A={1, 5} has exactly three proper subsets which are 𝜙, {1}, { 5}
or n(A)= 2 and number of proper subsets of A = (22) – 1 = 4 – 1 = 3
d. Exactly 4 subsets
Yes, A={1, 5} has exactly four subsets which are 𝜙, {1}, { 5}, {1, 5}
or n(A)= 2 and number of subsets of A = 22 = 4
e. Exactly 6 proper subsets
No, there is no set A such that n(A) = n and (2n) – 1 =6 which means (2n) =7
Since N
n
any
for
n
2
7
f. Exactly 30 subsets
No, for n(A) = n, then 2n
= 30 2
30
n
that
such
N
n
such
no
is
there
g. Exactly 14 proper subsets
No, if n(A) = n, then (2n
) – 1 = 14 2n
= 14+1=15 2
15
n
that
such
N
n
such
no
is
there
h. Exactly 15 proper subsets
Yes, A={1, 2, 3, 5} has exactly 15 proper subsets
Since n(A) = 4 (2n
) – 1 =1524
=15 +1 = 16 24
= 16 is true
10. How many elements does A contain if it has:
a. 64 subsets?
2n
=64 2n
= 26
n=6 is the number of elements of set A.
23. 23
b. 31 proper subsets?
(2n
) – 1 =31 2n
= 31+1=32 2n
= 25
n=5 is the number of elements of set A.
c. No proper subset?
(2n
) – 1 =0 2n
= 0+1=1 2n
= 20
n=0 is the number of elements of set A={ }.
e. 255 proper subsets?
(2n
) – 1 =255 2n
= 255+1=256 2n
= 28
n=8 is the number of elements of set A
11. Find the truth value of each of the following.
a. 𝜙∈(𝜙) True
b. 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑠𝑒𝑡 𝐴, 𝜙⊆(𝐴) True
c. 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑠𝑒𝑡 𝐴,∈𝑃(𝐴) True
d. 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑠𝑒𝑡 𝐴,⊂𝑃(𝐴). True
12. For any three sets 𝐴, 𝐵 and , prove that:
a. If 𝐴⊆𝐵 and 𝐵⊆𝐶, then 𝐴⊆𝐶.
C
x
A
x
x
C
x
B
x
x
B
x
A
x
x
b. If 𝐴⊂𝐵 and 𝐵⊂𝐶, then 𝐴⊂𝐶.
C
A
C
x
A
x
x
C
B
C
x
B
x
x
B
A
B
x
A
x
x
24. 24
Solutions/Answers to Exercises of pages 36-38
1. If 𝐵⊆𝐴, 𝐴∩𝐵′={1,4,5} and 𝐴∪𝐵={1,2,3,4,5,6}, find 𝐵.
n(𝐴∩𝐵′) = n(A) – n(B) = 3
n(A) = 3 + n(B)
since 𝐵⊆𝐴 then 𝐴∩𝐵 = B which means n(𝐴∩𝐵) = n(B)
n(𝐴∪𝐵) = 6
n(A) + n(B) = n(𝐴∪𝐵) + n(𝐴∩𝐵)
3 + n(B) + n(B) = 6 + n(B)
n(B) + n(B) - n(B) = 6 – 3
n(B) = 3
2. Let 𝐴={2,4,6,7,8,9}, 𝐵={1,3,5,6,10} and 𝐶={ 𝑥:3𝑥+6=0 or 2𝑥+6=0}. Find
a. 𝐴∪𝐵.
𝐴∪𝐵 = {2,4,6,7,8,9}∪{1,3,5,6,10}= ={1, 2. 3, 4, 5, 6, 7, 8, 9, 10}
b. Is (𝐴∪𝐵)∪𝐶=𝐴∪(𝐵∪𝐶)?
Yes which holds by associative property of union, ∪
3. Suppose 𝑈= The set of one digit numbers and
𝐴={𝑥: 𝑥 is an even natural number less than or equal to 9}
Describe each of the sets by complete listing method:
First let us determine U: U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and A: A = {2, 4, 6, 8}
Then:
a. 𝐴′. 𝐴′ = {0, 1, 3, 5, 7, 9}
b. 𝐴∩𝐴′. 𝐴∩𝐴′ = { } = 𝜙
c. 𝐴∪𝐴′. 𝐴∪𝐴′ = U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
d. (𝐴′)′ (𝐴′)′ = A = {2, 4, 6, 8}
e. 𝜙−𝑈. 𝜙−𝑈 = 𝜙
f. 𝜙′ 𝜙′ = 𝑈− 𝜙 = 𝑈 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
g. 𝑈′ 𝑈′ = 𝑈− 𝑈 =𝜙
25. 25
5. Use Venn diagram to illustrate the following statements:
a. (𝐴∪𝐵)′=𝐴′∩𝐵′.
b . (𝐴∩𝐵)′=𝐴′∪𝐵′.
d. If 𝐴⊈𝐵, then 𝐴𝐵≠𝜙.
26. 26
d.𝐴∪𝐴′=𝑈.
6. Let 𝐴={5,7,8,9} and 𝐶={6,7,8}. Then show that (𝐴𝐵)C=A(𝐵𝐶).
What is the operation between set A and (𝐵𝐶) and what is B ? [It is wrong question]
Anyway, let us start from (𝐴𝐵) and arrive at an equivalent conclusion using the properties
(𝐴𝐵)𝑐=(A∩B’) ∩C’ definition of relative complement of sets
A∩(B’∩C’)= A∩(B∪C)’ property of complement of ∪, union
A∩(B∪C)’= A(B∪C) definition of relative complement of sets
7. Perform each of the following operations.
a. 𝜙∩*𝜙} 𝜙∩*𝜙}= 𝜙
b. {𝜙,{𝜙}} – {{𝜙}} {𝜙,{𝜙}} – {{𝜙}}={𝜙}
c. {𝜙,{𝜙}} – {𝜙} {𝜙,{𝜙}} – {𝜙}={{𝜙}}
d. {{{𝜙}}} – 𝜙 {{{𝜙}}} – 𝜙={{{𝜙}}}
8. Let 𝑈 = {2,3,6,8,9,11,13,15}, 𝐴 = {𝑥|𝑥 is a positive prime factor of 66}
𝐵={ 𝑥∈𝑈| 𝑥 is composite number } and 𝐶 = {𝑥∈𝑈| 𝑥 – 5∈𝑈}. Then find each of the following.
𝐴∩𝐵, (𝐴∪𝐵)∩𝐶, (𝐴 – 𝐵)∪𝐶, (𝐴 – 𝐵) – 𝐶, 𝐴 – (𝐵 – 𝐶), (𝐴 – 𝐶) – (𝐵 – 𝐴), 𝐴’∩𝐵’∩𝐶’
Given
𝑈 = {2,3,6,8,9,11,13,15} 𝐴 = {2, 3, 11} 𝐵={ 6, 8, 9, 15 } 𝐶 = {8, 11, 13}
𝐴∩𝐵 = 𝜙 (𝐴∪𝐵)∩𝐶={ 2, 3, 6, 8, 9, 11, 15 }∩*8, 11, 13+ ={8, 11}
(𝐴 – 𝐵)∪𝐶 = {2, 3, 11}∪*8, 11, 13+={2, 3, 8, 11, 13}
(𝐴 – 𝐵) – 𝐶 = {2, 3, 11}– {8, 11, 13}= {2, 3}
𝐴 – (𝐵 – 𝐶) = 2, 3, 11} -{ 6, 9, 15 } ={2, 3, 11}
(𝐴 – 𝐶) – (𝐵 – 𝐴)= {2, 3} -={ 6, 8, 9, 15 } ={2, 3}
𝐴’∩𝐵’∩𝐶’ = *6, 8, 9, 13,15+∩*2, 3, 11,13+∩*2, 3, 6, 9, 15+= 𝜙
27. 27
9. Let 𝐴∪𝐵 = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑥, 𝑦, 𝑧} and 𝐴∩𝐵 = {𝑏, 𝑒, 𝑦}.
We recall the following: 𝐵 – 𝐴 = 𝐵∩𝐴’ = (𝐴∪𝐵) – A and 𝐴 – 𝐵 = 𝐴∩𝐵’ = (𝐴∪𝐵) – B
a. 𝐼𝑓 𝐵 – 𝐴 = {𝑥, 𝑧}, then 𝐴 = ________________________
𝐴 = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑦 }
b. 𝐼𝑓 𝐴 – 𝐵 =𝜙, then 𝐵 = _________________________
B = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑥, 𝑦, 𝑧}
c. 𝐼𝑓 𝐵 = {𝑏, 𝑒, 𝑦, 𝑧}, then 𝐴 – 𝐵 = ____________________
𝐴 – 𝐵 = {𝑎, 𝑐, 𝑑, 𝑥}
10. Let 𝑈 = {1, 2, …, 10}, 𝐴 ={3, 5, 6, 8, 10}, 𝐵 = {1, 2, 4, 5, 8, 9}, 𝐶 = {1, 2, 3, 4, 5, 6, 8} and
𝐷 = {2, 3, 5, 7, 8, 9}.
Verify each of the following.
a. (𝐴∪𝐵)∪𝐶 = 𝐴∪(𝐵∪𝐶).
(𝐴∪𝐵)∪𝐶 = 𝐴∪(𝐵∪𝐶)={ {1, 2, 3, 4, 5, 6, 8, 9, 10}
[Associative property of ∪]
b. 𝐴∩(𝐵∪𝐶∪𝐷) = (𝐴∩𝐵)∪(𝐴∩𝐶)∪(𝐴∩𝐷)= {3,5,6,8}
[Distributive property of ∩ over ∪]
c. (𝐴∩𝐵∩𝐶∩𝐷)’= 𝐴’∪𝐵’∪𝐶’∪𝐷’ = {5, 8}
[Distributive property of absolute complement over ∩]
d. 𝐶 – 𝐷 = 𝐶∩𝐷’ = {1, 4, 6}
[Property relating relative complement and absolute complement]
e. 𝐴∩(𝐵∩𝐶)’= (𝐴 – 𝐵)∪(𝐴 – 𝐶) ={3, 6, 10}
[Property relating distributive property of absolute complement over ∩ and relative complement]
11. Depending on question No. 10 find.
a. 𝐴 Δ 𝐵
𝐴 Δ 𝐵 = (𝐴 – 𝐵)∪(B – A)= {1, 2, 3, 4, 6, 9, 10 }
b. 𝐶 Δ 𝐷 =(C – D)∪(D – C) = {1, 4, 6, 7}
c. (𝐴 Δ 𝐶)Δ 𝐷
Let M= 𝐴 Δ 𝐶 = (A – C ) ∪(C – A ) = {1, 2, 4, 10} D = 𝐷 = {2, 3, 5, 7, 8, 9}.
Then (𝐴 Δ 𝐶)Δ 𝐷 = M Δ 𝐷= (M – D ) ∪ (D – M ) = {1, 3, 4, 5, 7, 8, 9, 10}
Therefore, (𝐴 Δ 𝐶)Δ 𝐷 = {1, 3, 4, 5, 7, 8, 9, 10}
28. 28
d. (𝐴∪𝐵) (𝐴 Δ 𝐵)
𝐴∪𝐵) (𝐴 Δ 𝐵) = (𝐴∪𝐵) ∩ (𝐴 Δ 𝐵)’= (𝐴∪𝐵) ∩ [(𝐴 ∪𝐵)-(A∩B)]’= (𝐴∪𝐵) ∩ [(𝐴 ∪𝐵) ∩ (A∩B)’]’
(𝐴∪𝐵) ∩ [(𝐴 ∪𝐵)’ ∪ (A∩B)] = [(𝐴∪𝐵) ∩(𝐴 ∪𝐵)’] ∪[(𝐴∪𝐵) ∩(A∩B)]= 𝜙 ∪(A∩B)
= A∩B = {5, 8}
12. For any two subsets 𝐴 and 𝐵 of a universal set 𝑈, prove that:
a. Δ 𝐵 = 𝐵 Δ 𝐴.
𝐴 Δ 𝐵 = (A – B ) ∪(B – A ) = (B – A ) ∪ (A – B ) = 𝐵 Δ 𝐴
b. 𝐴 Δ 𝐵 = (𝐴∪𝐵) – (𝐴∩𝐵)= (𝐵 ∪𝐴) – (𝐵 ∩𝐴)= 𝐵 Δ 𝐴 .
𝐴 Δ 𝐵 = (A – B ) ∪(B – A ) = (A ∩ B’ ) ∪(B ∩ A’ ) = (𝐴∪𝐵) ∩ (𝐴∩𝐵)’= (𝐴∪𝐵) – (𝐴∩𝐵)
c. 𝐴 Δ 𝜙= 𝐴.
𝐴 Δ 𝜙= 𝐴= (𝐴∪ 𝜙) – (𝐴∩ 𝜙)= 𝐴– 𝜙= 𝐴∩ 𝜙’= 𝐴∩ U=A
d. 𝐴 Δ 𝐴 =𝜙
𝐴 Δ 𝐴 =(𝐴∪ A) – (𝐴∩ A)= 𝐴– A= 𝐴∩ A’= 𝜙
13. Draw an appropriate Venn diagram to depict each of the following sets.
a. U = The set of high school students in Addis Ababa.
A = The set of female high school students in Addis Ababa.
B = The set of high school anti-AIDS club member students in Addis Ababa.
C = The set of high school Nature Club member students in Addis Ababa.
29. 29
b. U = The set of integers.
A = The set of even integers.
B = The set of odd integers.
C = The set of multiples of 3.
D = The set of prime numbers.
30. 30
Unit TWO
Solutions/Answers to Exercises 2.1 of page 47
1. Find an odd natural number x such that LCM (x, 40) = 1400.
Solution:- x is odd means x = 2n+1, N
n
Let GCF(x, 40) = g
Therefore, 1400g= 40x = 40(2n+1)
)
n
(
x
n
x
n
x
1
2
35
1
2
35
1
2
40
1400
When n=1, x=35(2x1+1) = 35x3 = 105 which is odd
When n=2, x=35(2x2+1) = 35x5 = 175 which is odd
When n=1, x=35(2x3+1) = 35x7 = 245 which is odd
.
.
.
In a similar way, the odd natural number x such that LCM (X, 40) = 1400 is the number
X = )
n
( 1
2
35 , N
n
2. There are between 50 and 60 number of eggs in a basket. When Loza counts by 3’s, there are 2 eggs left over.
When she counts by 5’s, there are 4 left over. How many eggs are there in the basket?
Solution:- Let the number of eggs be n, 50 ≤n ≤60 , x & y are number of counts
Then n = 3x+2 = 5y+4
i.e., n = 3x+2 50 ≤3x+2 ≤6048 ≤3x ≤5848/3 ≤x≤58/316 ≤x+ ≤19.333 …16 ≤x+ ≤19
n= 5y+450 ≤5y+4 ≤6046 ≤5y ≤5646/5≤y ≤56/546/5≤y ≤56/59.2≤y ≤11.29≤y ≤11
3x+2=5y+4 3x-5y=4-23x-5y=2
When x=16, 3(16)-5y=248-2=5y46=5y46/5=y not counting number
Proceeding in a similar way, when x=19, 3(19)-5y=257-2=5y55=5y55/5=y=11
Therefore, there are n = 3x+2=3(19)+2=59 eggs
3. The GCF of two numbers is 3 and their LCM is 180. If one of the numbers is 45, then find the second number.
Solution:- Let the second number be y, then 3x180=45y y= 540/45=12 y=12
4. Using Mathematical Induction, prove the following:
a) 6n
- 1 is divisible by 5 for n ≥0
Proof:-
(1) For n=0, 60
- 1=1-1=0 is divisible by 5 is true and
31. 31
for n=1, 61
- 1=6-1=5 is divisible by 5 is true
(2) Assume for n=k, 6k
- 1 is divisible by 5 is true i.e., m𝐖 such that 6k
- 1= 5m
We should show that it is true for n = k+1,
Claim:- 6k+1
- 1 is divisible by 5 or d𝐖 such that 6k+1
- 1= 5d
Now 6k
- 1= 5m 6k
= 5m +1
6. (6k
)= 6(5m+1) = 6(5m) +6
(6k+1
) – 1 =6(5m) +6 – 1
(6k+1
) – 1 =6(5m) +5
(6k+1
) – 1 =5[(6m) +1], m, 6m𝐖, (6m+1)𝐖
(6k+1
) – 1 =5d, where d=[(6m) +1]𝐖 is true
Therefore, 6n
- 1 is divisible by 5 for any n ≥0
(3) b) 2n
≤ (n+1)! , for n ≥ 0
Proof:- For n=0, 20
≤ (1+0)! 1≤ 1! 1≤ 1 is true and
for n=1, 21
≤ (1+1) ! 2≤ 2! 2≤ 2is true
Assume that it is true for n = k i.e., 2k
≤ (k+1)!
We should show that it is true for n = k+1
Claim:- 2(k+1)
≤ [(k+1) +1]!= [(k+2)!
Now, 2k
≤ (k+1)! 2k
(2 )≤ 2(k+1)! ≤ (k+2)(k+1)! Since 2 ≤ k+1 ≤ k+2
2k
(2 )= 2k+1
≤ (k+2)(k+1)! = (k+2)! 2k+1
≤ (k+2)!
c) xn
+ yn
is divisible by x+y , for odd natural number n ≥ 1
Proof:- For n=1, x1
+ y1
is divisible by x+y is true
Assume that it is true for an odd number n = k
i.e., xk
+yk
is divisible by x+y is true ; since k is odd, m𝐖 such that k = 2m+1
xk
+yk
= t(x+y) , for some N
t , is true i.e., x2m+1
+y2m+1
= t(x+y) , for some N
t , is true
We should show that it is true for the next odd number n = k+2=2m+1+2=2(m+1)+1
Claim:- x(k+2)
+y(k+2)
= x(2m+2+1)
+y(2m+2+1)
is divisible by x+y is true
Or x(2m+2+1)
+y(2m+2+1)
= x2(m+1)+1)
+y2(m+1)+1)
= d(x+y), for some N
d
x2m+1
+y2m+1
= t(x+y)
(x)(x2m
) +(y)(y2m
) = (x)(x2m
)+ (y)(x2m
) -(y)(x2m
) +(y)(y2m
) +(x)(y2m
) -(x)(y2m
)
= (x)(x2m
)+ (y)(x2m
) -(y)(x2m
) +(y)(y2m
) +(x)(y2m
) -(x)(y2m
)
= (x2m
)(x+ y) -(y)(x2m
) +(y2m
) (y+x) -(x)(y2m
)
=(x2m
)(x+ y) +(y2m
) (y+x) -(y)(x2m
) -(x)(y2m
)
= (x+ y) ( x2m
+y2m
) - (y)(x2m
) -(x)(y2m
) = (x+ y) ( x2m
+y2m
) - (xy)(x2m-1
) -(xy)(y2m-1
)
32. 32
Here = (x+ y) ( x2m
+y2m
) -(xy)[(x2m-1
) +(y2m-1
)] x
= (x+ y) ( x2m
+y2m
) -(xy)[k(x+ y)] because (x2m-1
) +(y2m-1
) & x2m
+y2m
are is divisible by x+y
(x2m-1
) +(y2m-1
) = k(x+ y) and x2m
+y2m
= (x2m
) +(y2m
) = r(x+ y) where k, rN
(x+ y) ( x2m
+y2m
) -(xy)[k(x+ y)] == (x+ y) r(x+ y) -(xy)[k(x+ y)] =(x+ y)[ r(x+ y) - k(xy)]
which is divisible by x+y since {(x+ y)[ r(x+ y) - k(xy)]}/(x+y) = r(x+ y) - k(xy) We are done
d) 2+4+6+ … + 2n = n(n+1)
Proof:- For n=1, 2 = 1(1+1)! 2= 2 is true
Assume that it is true for n = k i.e., 2+4+6+ … + 2k = k(k+1) is true
We should show that it is true for n = k+1
Claim:- 2+4+6+ … + 2(k+1) = (k+1)(k+1+1) is true
Now, 2+4+6+ … + 2k = k(k+1)
2+4+6+ … + 2k +2(k+1)= k(k+1)+2(k+1)
2+4+6+ … + 2k +2(k+1)= (k+1) (k+2)
Therefore, 2+4+6+ … +2(k+1)= (k+1) (k+2) is true
Alternative Way
1
2
1
n
n
i
n
i
Proof:- For n=1, 2
2
1
1
1
1
2
1
1
i
x is true
Assume that it is true for n = k i.e.,
1
2
1
k
k
k
k
i
is true
We should show that it is true for n = k+1
Claim:-
2
1
2
1
1
k
k
k
k
i
is true
Now,
1
2
1
k
k
k
k
i
)
k
(
k
k
)
k
(
k
k
i
1
2
1
1
2
2
1
)
k
(
k
k
k
k
i
1
2
1
2
1
1
)
k
(
k
k
k
i
2
1
2
1
1
Therefore, )
k
(
k
k
k
i
2
1
2
1
1
is true (We are done)
Divisible by x+y Divisible by x+y, since odd number2m-1<k
33. 33
e)
6
1
2
1
3
2
1 2
2
2
2 )
n
)(
n
(
n
n
...
Proof:- For n=1, 12
= 1
6
6
6
1
1
2
1
1
1
)
x
)(
(
is true
Assume that it is true for n = k i.e., 12
+22
+32
+ … + k2
=
6
1
2
1 )
k
)(
k
(
k
is true is true
We should show that it is true for n = k+1
Claim:-
12
+22
+32
+ … + k2
+ (k+1)2
=
6
1
1
2
1
1
1
k
)
k
(
k
=
6
3
2
2
1
k
)
k
(
k
is true
Now, [11
+22
+32
+ … + k2
] + (k+1)2
=
6
1
2
1 )
k
)(
k
(
k
+ (k+1)2
11
+22
+32
+ … + k2
+ (k+1)2
=
6
6
1
2
1 2
1)
+
(k
)
k
)(
k
(
k
11
+22
+32
+ … + k2
+ (k+1)2
=
6
1
6
1
2
1
k
)
k
(
k
)
k
(
11
+22
+32
+ … + k2
+ (k+1)2
=
6
6
7
2
1 2
k
k
)
k
(
11
+22
+32
+ … + k2
+ (k+1)2
=
6
6
7
2
1 2
k
k
)
k
(
11
+22
+32
+ … + k2
+ (k+1)2
=
6
3
2
2
1
k
k
)
k
(
Therefore, 11
+22
+32
+ … + k2
+ (k+1)2
= =
6
3
2
2
1
k
)
k
(
k
is true
Alternative Way
6
1
2
1
1
2
n
n
n
i
n
i
Proof:- For n=1,
6
1
1
2
1
1
1
1
1
2
x
n
i
For n=1,
1
6
6
6
3
2
6
1
1
2
1
1
1
1
x
is true
Assume that it is true for n = k i.e.,
6
1
2
1
1
2
k
k
k
i
k
i
is true
34. 34
We should show that it is true for n = k+1
Claim:-
6
1
1
2
1
1
1
1
1
2
k
k
k
i
k
i
is true
Simplifying RHS
6
3
2
2
1
1
1
2
k
k
k
i
k
i
is true
Now,
6
1
2
1
1
2
k
k
k
i
k
i
2
2
1
2
1
6
1
2
1
1
k
k
k
k
k
i
k
i
6
1
6
1
2
1
1
2
2
1
2
k
k
k
k
k
i
k
i
6
1
6
1
2
1
1
2
2
1
2
k
k
k
k
k
i
k
i
6
1
6
1
2
1
1
2
1
2
k
k
k
k
k
i
k
i
2
1
2
1
k
i
k
i
6
3
2
2
1
k
k
)
k
(
We are done
f)
4
1
3
2
1
2
2
3
3
3
3 )
n
(
n
n
...
Proof:- For n=1, 13
= 1
4
4
1
4
1
1
1
1
2
2
3
)
(
is true
Assume that it is true for n = k i.e., 13
+23
+33
+ … + k3
=
4
1 2
2
)
n
(
n
is true is true
We should show that it is true for n = k+1
Claim:- 13
+23
+33
+ … + k3
+ (k+1)3
=
4
1
1
1 2
2
)
k
(
k
=
4
2
1 2
2
)
k
(
k
is true
Now, [13
+23
+33
+ … + k3
] + (k+1)3
=
4
1 2
2
)
k
(
)
k
(
+ (k+1)3
13
+23
+33
+ … + k3
+ (k+1)3
=
4
4
1 3
2
2
1)
+
(k
)
k
(
k
35. 35
13
+23
+33
+ … + k3
+ (k+1)3
=
4
1
4
1 2
2
k
k
)
k
(
13
+23
+33
+ … + k3
+ (k+1)3
=
4
4
4
1 2
2
k
k
)
k
(
11
+22
+32
+ … + k2
+ (k+1)2
=
4
2
1
2
2
k
)
k
(
11
+22
+32
+ … + k2
+ (k+1)2
=
6
3
2
2
1
k
k
)
k
(
Therefore, 13
+23
+33
+ … + k3
+ (k+1)3
=
4
2
1 2
2
)
k
(
k
is true
Alternative Way
6
1
2
1
1
2
n
n
n
i
n
i
Proof:- For n=1,
6
1
1
2
1
1
1
1
1
2
x
n
i
For n=1,
1
6
6
6
3
2
6
1
1
2
1
1
1
1
x
is true
Assume that it is true for n = k i.e.,
6
1
2
1
1
2
k
k
k
i
k
i
is true
We should show that it is true for n = k+1
Claim:-
6
1
1
2
1
1
1
1
1
2
k
k
k
i
k
i
is true
Simplifying RHS
6
3
2
2
1
1
1
2
k
k
k
i
k
i
is true
Now,
6
1
2
1
1
2
k
k
k
i
k
i
2
2
1
2
1
6
1
2
1
1
k
k
k
k
k
i
k
i
6
1
6
1
2
1
1
2
2
1
2
k
k
k
k
k
i
k
i
6
1
6
1
2
1
1
2
2
1
2
k
k
k
k
k
i
k
i
36. 36
6
1
6
1
2
1
1
2
1
2
k
k
k
k
k
i
k
i
2
1
2
1
k
i
k
i
6
3
2
2
1
k
k
)
k
(
We are done
g)
1
1
1
4
3
1
3
2
1
2
1
1
n
n
)
n
(
n
...
Proof:- For n=1,
2
1
2
1
1
2
1
2
1
1
is true
Assume that it is true for n = k i.e.,
1
1
1
4
3
1
3
2
1
2
1
1
k
k
)
k
(
k
... is true is true
We should show that it is true for n = k+1
Claim:-
1
1
1
2
1
1
1
1
4
3
1
3
2
1
2
1
1
k
k
k
k
)
k
(
k
... is true
Now,
1
1
1
4
3
1
3
2
1
2
1
1
k
k
)
k
(
k
...
2
1
1
1
2
1
1
1
1
4
3
1
3
2
1
2
1
1
k
k
k
k
k
k
)
k
(
k
...
2
1
1
2
2
1
1
1
1
4
3
1
3
2
1
2
1
1
k
k
)
k
(
k
k
k
)
k
(
k
...
2
1
1
2
2
1
1
1
1
4
3
1
3
2
1
2
1
1 2
k
k
k
k
k
k
)
k
(
k
...
2
1
1
2
1
1
1
1
4
3
1
3
2
1
2
1
1
2
k
k
k
k
k
)
k
(
k
...
2
1
2
1
1
1
1
4
3
1
3
2
1
2
1
1
k
k
k
k
)
k
(
k
...
1
1
1
2
1
1
1
1
4
3
1
3
2
1
2
1
1
k
k
k
k
)
k
(
k
... We are done
37. 37
Solutions/Answers to Exercises 2.2 of page 55
2. Express each of the following rational numbers as decimal:
a)
9
4
= 0.444 … = 4
0.
b)
25
2
= 0.08
c)
7
11
= 1.571428571428 … = 571428
1.
d)
3
2
5
= 6
5.
e)
77
2
= 0.025974025974 … 025974
.
0
2. Write each of the following as decimal and then as a fraction:
a) three tenths =
10
3
3
0
.
b) four thousands = 4,000
3. Write each of the following in meters as a fraction and then as a decimal
a) 4 mm =
1000
4
m = 0.004 m
b) 6 cm and 4 mm =
1000
64
m = 0.064 m
c) 56 cm and 4 mm =
1000
564
m = 0.564 m
4. Classify each of the following as terminating or non-terminating periodic
a)
13
5
it is non-terminating periodic
since the denominator has the prime factor, 13, different from 2 and 5
in its lowest term
b)
10
7
terminating
since the only prime factors of the denominator are 2 and 5 in its lowest term
c)
64
69
terminating
since the only prime factors of the denominator is 2 in its lowest term
d)
60
11
it is non-terminating periodic
since the denominator has the prime factor, 3, different from 2 and 5
in its lowest term
e)
12
5
it is non-terminating periodic
since the denominator has the prime factor, 3, different from 2 and 5
in its lowest term
38. 38
5. Convert the following decimals to fractions:
a)
90
293
90
32
325
90
32
325
90
5
32
5
325
10
100
10
5
2
3
100
5
2
3
10
100
10
100
5
2
3
10
10
10
10
5
2
3
5
2
3 1
1
1
1
1
1
.
.
x
.
x
.
.
.
.
b) 10
0
100
10
14
3
0
1000
14
3
0
10
1000
10
1000
14
3
0
10
10
10
2
10
14
3
0
14
3
0 1
2
1
1
1
1
x
.
x
.
.
.
.
990
311
990
3
314
990
14
3
14
314
.
.
c)
999
275
999
275
0
275
275
1
1000
1
275
0
1000
275
0
1
1000
1
1000
275
0
10
10
10
10
275
0
275
0 0
3
0
0
3
0
.
.
x
.
x
.
.
.
.
6. Determine whether the following are rational or irrational:
a) 5
7
2. is rational number which is non-terminating periodic number
b) 0.272727 … is rational number which is non-terminating periodic number which is 27
0.
c)
2
1
8 is irrational number which is neither terminating nor
non-terminating periodic number which is equal to:
...
.
...
.
.
.
12132034
2
4142135
1
5
1
2
5
1
2
2
3
2
3
2
1
4
2
1
16
2
1
8
2
2
1
8
7. Which of the following statements are true and which of them are false?
a) The sum of any two rational numbers is rational
True since the set of rational is closed under addition
b) The sum of any two irrational numbers is irrational
False since for irrational number m and m+ –m =0 is rational number
c) The product of any two rational numbers is rational
True since the set of rational is closed under multiplication
d) The product of any two irrational numbers is irrational
False since for irrational number 2
2
2
2
, is rational number
11. Find two rational numbers between
3
1 and
2
1
2
2
1
3
1
=
12
5
2
6
3
2
and
2
1
2
1
3
2
3
1
=
36
17
36
9
8
4
1
9
2
Note for any two rational numbers m & n : 0.5m+0.5n
and 0.6m+0.5n are rational numbers between m & n
39. 39
Solutions/Answers to Exercises 2.3 of page 60-61
1. Verify that
a. i
i
i
i 2
2
1
2
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
2
2
0
2
2
2
2
1
2
2
2
2
2
2
1
2 2
b. (2, -3)(-2, 1) = (-1, 8)
8
,
1
8
1
3
8
4
)
1
(
3
6
2
4
3
2
3
2
2
2
2
3
2
1
,
2
3
,
2
i
i
i
i
i
i
i
i
i
i
c.
1
,
2
10
1
,
5
1
1
,
3
1
3,
1
,
2
2
10
10
5
10
10
1
5
1
10
10
1
5
1
1
9
10
1
5
1
)
1
(
3
3
9
3
3
3
3
10
1
5
1
3
3
1
,
2
10
1
,
5
1
1
,
3
1
3,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
d. i
i 9
1
6
3
3i
2
2
i
i
i
i
i
i
i
i
i
9
1
3
12
9
6
4
6
3
)
1
(
9
12
4
6
3
3
3
2
2
4
6
3
3i
2
2
2
2. Show that
a. )
Im(
)
Re( z
iz
Let z = x + yi
y
yi
x
yi
x
z
and
y
y
xi
i
y
xi
yi
x
i
iz
)
Im(
)
Im(
)
Im(
)
Re(
)
)
(
Re(
))
(
Re(
)
Re( 2
Therefore, y
z
iz
)
Im(
)
Re(
b.
)
Re(
)
Im( z
iz
Let z = x + yi
x
yi
x
z
x
y
xi
i
y
xi
i
y
xi
yi
x
i
iz
)
Re(
)
Re(
)
Im(
Im
Im
)
(
Im
)
Im(
2
2
c. 1
2
1 2
2
z
z
z Let z = x + yi
i
y
xy
y
x
x
y
yi
xyi
x
x
yi
yi
x
x
yi
x
yi
x
z
2
2
1
2
2
2
1
2
1
2
1
1
1
1
2
2
2
2
2
2
2
2
2
i
y
xy
y
x
x
yi
x
y
xyi
x
yi
x
yi
xyi
x
yi
x
yi
x
z
z
2
2
1
2
1
2
2
2
1
2
2
2
1
2
1
2
2
2
2
2
2
2
2
2
i
y
xy
y
x
x
z
z
z
Therefore 2
2
1
2
1
2
1
, 2
2
2
2
40. 40
3. Do the following operations and simplify your answer.
a)
i
i
i
i
5
2
4
3
2
1
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
25
2
125
58
625
50
625
290
625
90
50
200
225
400
90
120
150
200
15
20
15
20
15
20
6
10
20
15
10
6
20
15
4
6
12
8
3
5
20
15
4
8
3
6
12
5
5
4
15
4
8
3
6
2
6
5
5
4
3
2
4
3
2
1
5
5
2
4
3
2
1
2
b)
)
3
(
)
2
(
)
1
(
5
i
i
i
i
2
1
10
5
3
10
3
5
3
9
3
5
3
3
1
5
3
1
3
2
5
3
2
2
5
3
2
1
5
2
2
i
i
i
i
i
i
i
i
i
i
i
i
i
i
)
i
(
i
i
i
i
)
i
(
)
i
(
)
i
(
i
c) 3
1 i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
2
2
2
2
2
2
1
2
1
1
2
1
1
1
1
1
1
1
2
2
3
4. Locate the complex numbers z1+z2 and z1-z2, as vectors where
a) i
z 2
1 , i
z
3
2
2
i
i
i
z
z
3
2
3
2
2
2
1
i
i
i
i
i
i
z
z 3
3
2
3
2
3
3
2
2
3
2
2
2
1
b)
0
3
1
3 2
1 ,
z
,
,
z
1
0
0
1
3
3
0
3
1
3
2
1 ,
,
,
,
z
z
1
3
2
0
1
3
3
0
3
1
3
2
1 ,
,
,
,
z
z
c)
4
1
1
3 2
1 ,
z
,
,
z
5
2
4
1
1
3
4
1
1
3
2
1 ,
,
,
,
z
z
3
4
4
1
1
3
4
1
1
3
2
1
,
,
,
,
z
z
41. 41
d) ib
a
z
,
bi
a
z
2
1
0
2
1 ,
a
oi
a
bi
bi
a
a
ib
a
bi
a
z
z
b
,
bi
bi
bi
a
a
ib
a
bi
a
z
z 2
0
2
0
2
1
5. Sketch the following set of points determined by the condition given below:
a) 1
i
+
1
-
Z
Let z = x+yi
1
1
1
1
1
1
1
1
2
2
2
2
y
1
-
x
y
1
-
x
i
1
y
1
-
x
i
+
yi
1
-
x
i
+
1
-
yi
x
i
+
1
-
Z
Which represents points on the circle with center (1, 1) and radius 1.
b) 3
i
z
Let z = x+yi
9
1
3
1
3
1
3
3
2
2
2
2
y
x
y
x
i
y
x
i
yi
x
i
z
Which represents points inside the circle with center (0, -1) and radius 3
c) 4
4
i
z
Let z = x+yi
16
4
4
4
4
4
4
4
4
4
2
2
2
2
y
x
y
x
i
y
x
i
yi
x
i
z
Which represents points outside the circle with center (0, 4) and radius 4
6. Using properties of conjugate and modulus, show that
a) i
z
i
z 3
3
i
z
i
z
i
z
i
z
i
z 3
3
0
3
0
3
3
b) z
i
iz
z
i
z
i
z
i
z
i
iz
0
0
c) i
i 4
3
2
2
i
i
i
i
i
i 4
3
4
3
1
4
4
4
4
2 2
2
7. Show that
i
i
1
8
1
7
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
8
8
8
8
8
8
1
8
1
8
1
2
1
1
2
1
1
1
1
2
3
3
3
2
7
8. Using mathematical induction, show that (when n = 2, 3, . . . ,)
n
n z
...
z
z
z
...
z
z
2
1
2
1
For n=1 : 1
1 z
z is true
For n=2: 2
1
2
1 z
z
z
z
by property of congugate
Assume, it is true for n=k , i.e., k
k z
...
z
z
z
...
z
z
2
1
2
1
We want to prove that it is true for n=k+1 , that 1
2
1
1
2
1
k
k
k
k z
z
...
z
z
z
z
...
z
z
Let k
z
z
z
M
...
2
1 . Then k
z
z
z
M
...
2
1
42. 42
And 1
1
K
k z
M
z
M
1
1
2
1 ... K
k
k z
M
z
z
z
z 1
2
1 ...
k
k z
z
z
z
Hence n
n z
z
z
z
z
z
...
... 2
1
2
1
9. Show that the equation r
z
z
0 which is a circle of radius r centered at o
z can be written as
2
2
0
2
Re
2 r
z
z
z
z
Correction:-The question must be corrected as 2
2
0
0
2
Re
2 r
z
z
z
z
Let
0
0
0 ,
, y
x
z
and
y
x
z
2
0
2
0
2
0
2
0
2
0
2
0
2
0
2
0
2
0
2
0
2
0
2
0
2
0
0
0
0
0
0
, y
x
z
y
y
x
x
y
y
x
x
y
y
x
x
y
y
x
x
y
y
x
x
r
i
y
y
x
x
i
y
x
yi
x
r
z
z
2
0
2
0
0
0
2
2
2
0
0
2
2
0
0
2
2
0
2
0
2
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
2
2
2
2
2
2
2
,
2
2
2
2
2
2
Re
Re
2
,
y
x
yy
xx
y
x
y
yy
y
x
xx
x
y
y
x
x
r
yy
xx
yy
yi
x
i
xy
xx
z
z
yy
yi
x
i
xy
xx
i
yy
yi
x
i
xy
xx
i
y
x
yi
x
z
z
y
x
z
And
done
are
we
that
means
which
same
the
y
x
yy
xx
y
x
r
And
y
x
yy
xx
y
x
z
z
z
z
Now
2
0
2
0
0
0
2
2
2
2
0
2
0
0
0
2
2
2
0
0
2
2
2
2
2
Re
2
43. 43
Solutions/Answers to Exercises 2.4 of page 67
1. Find the argument of the following complex numbers:
i
i
z
a
1
3
)
2
3
2
3
2
3
3
1
3
3
1
1
1
3
2
2
i
i
i
i
i
i
i
i
i
4
)
1
arg(
2
/
3
2
/
3
arg
arg
z
6
3
) i
z
b
6
/
6
3
1
tan
6
3
arg
6
3
arg
)
arg(
, 1
6
i
i
z
Therefore
2. Show that:
i
i
e
e
b
)
sin
cos
,
sin
cos
sin
cos
1
sin
cos
sin
cos
1
,
1
,
1
,
1
,
1
i
e
e
Therefore
i
i
e
and
i
e
i
e
n
r
e
from
and
n
r
e
From
i
i
i
i
i
i
i
3. Using Mathematical induction, show that:
n
n
i
i
i
i
e
e
e
e
...
2
2
1
.
.
. ,n=2, 3, …
Proof:-
True
is
e
i
i
i
e
e
n
For
i
i
i 2
2
1
2
1
2
1
2
2
1
1 sin
cos
sin
cos
sin
cos
.
,
2
k
k
i
i
i
i
e
e
e
e
e
i
k
n
for
True
is
it
Assume
...
2
2
1
.
.
.
.,
.
,
True
is
e
e
e
e
e
e
i
i
i
e
e
e
e
e
e
e
e
e
e
e
e
e
i
k
n
for
True
is
it
that
prove
to
wan
We
n
k
k
k
k
k
k
k
k
k
i
i
i
i
i
i
k
k
k
k
k
k
k
k
i
i
i
i
i
i
i
i
i
i
i
i
...
...
1
2
1
1
2
1
1
1
2
1
2
1
...
...
2
1
2
1
1
2
1
2
1
2
2
1
1
2
1
2
1
.
.
.
.
...
sin
...
cos
sin
cos
...
sin
...
cos
.
.
.
.
.
.
.
.
.
.
.,
.
,
1
1
)
i
e
a
Note :-
n
i
n
n
n
n
e
r
n
i
n
r
z
and
eger
an
is
k
k
z
n
z
sin
cos
int
,
2
arg
arg
1
1
sin
cos
sin
cos
,
1
,
1
,
2
2
i
e
and
n
r
e
From i
i
44. 44
4. show that:
2
3
sin
cos
3
cos
3
cos
)
a
2
3
2
2
3
2
2
sin
cos
3
cos
cos
sin
2
cos
sin
cos
sin
cos
sin
2
cos
sin
cos
sin
2
sin
cos
2
cos
2
cos
3
cos
)
a
5. show that:
1
,
1
1
...
1
1
2
z
for
z
z
z
z
z
n
n
1
,
1
1
1
...
...
1
1
1
...
1
...
1
1
1
2
2
2
2
z
for
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
n
n
n
n
n
n
6. Find the square root of:
i
z 9
1
,
0
,
2
,
2
,
9
9
k
n
r
i
z
2
,
2
,
sin
3
9 4
2
2
2
1
2
1
n
ce
e
e
e
r
C
k
i
k
i
n
n
k
n
i
n
k
2
2
3
2
2
3
2
2
2
2
3
4
sin
4
cos
3
3
,
0 4
0
i
i
i
e
C
k
If
i
2
2
3
2
2
3
2
2
2
2
3
4
5
sin
4
5
cos
3
3
3
,
1 4
5
4
1
i
i
i
e
e
C
k
If
i
i
i
C
and
i
C
are
i
z
of
roots
square
the
Therefore
2
2
3
2
2
3
2
2
3
2
2
3
:
9
, 1
0
7. Find the cube root of:
i
z 8
2
,
1
,
0
,
3
,
2
3
,
9
8
k
n
r
i
z
3
,
2
3
,
sin
2
8 3
2
2
3
2
3
1
2
1
n
ce
e
e
e
r
C
k
i
k
i
n
n
k
n
i
n
k
45. 45
i
i
i
e
C
k
If
i
2
0
2
2
sin
2
cos
2
2
,
0 2
0
i
i
i
e
e
C
k
If
i
i
2
3
2
1
2
3
2
6
7
sin
6
7
cos
2
2
2
,
1 6
7
3
2
2
1
i
i
i
e
e
C
k
If
i
i
2
3
2
1
2
3
2
6
11
sin
6
11
cos
2
2
2
,
2 6
11
3
4
2
2
i
C
and
i
C
i
C
are
i
z
of
roots
cube
the
Therefore
2
3
2
3
,
2
:
8
,
2
1
0
8. Solve the following equations:
i
z
a 8
) 2
/
3
4
,
0
64
,
0
64
0
3
,
0
3
0
64
3
3
0
64
3
0
3
0
64
3
0
3
64
3
0
3
64
3
64
3
3
3
3
8
8
3
2
3
2
3
2
2
2
3
2
2
2
3
2
2
2
3
3
2
2
3
3
2
2
3
3
2
2
3
3
2
3
2
3
x
y
x
y
x
x
y
y
x
or
y
and
xy
x
y
x
or
y
and
xy
x
y
x
or
y
and
xy
x
y
x
y
and
xy
x
y
y
x
and
xy
x
i
y
y
x
xy
x
i
y
xy
yi
x
x
z
i
yi
x
i
z
yi
x
z
Let
2
,
3
2
2
,
3
8
,
3
64
8
,
3
64
8
,
3
64
9
,
3
64
3
3
,
3
3
3
3
3
3
2
3
x
y
x
x
y
x
x
y
x
x
y
x
x
y
x
x
x
y
x
x
x
x
y
2
,
3
2
2
,
3
8
,
3
64
8
,
3
64
8
,
3
64
9
,
3
64
3
3
,
3
3
3
3
3
3
2
3
x
y
x
x
y
x
x
y
x
x
y
x
x
y
x
x
x
y
x
x
x
x
y
3
2
,
2
,
3
2
,
2
,
0
,
4
.
.
S
S
0
4
) 2
i
z
b
4
,
0
x
y
y
x
or
y
3
0
2
,
2
,
2
,
2
.
.
2
2
2
0
4
2
0
4
2
0
4
2
0
/
2
/
2
0
2
0
4
2
0
4
2
4
2
4
2
4
0
4
4
2
4
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
S
S
y
and
x
x
y
and
x
x
y
and
x
x
x
y
and
x
x
x
y
and
x
x
x
y
and
y
x
xy
and
y
x
xy
and
y
x
i
xyi
y
x
i
xyi
y
x
i
y
xyi
x
i
yi
x
i
z
and
yi
x
z
Let
46. 46
0
4
) 2
i
z
c
2
,
2
,
2
,
2
,
2
,
2
,
2
,
2
.
.
2
2
,
2
2
2
,
2
2
2
4
2
2
,
2
2
2
4
2
4
2
4
2
0
4
2
4
2
4
4
2
2
2
2
2
2
2
2
2
2
2
2
S
S
y
and
x
y
and
x
y
x
y
x
x
and
y
x
x
and
y
x
x
and
y
x
Again
y
x
y
x
x
and
y
x
x
and
y
x
x
and
y
x
xy
and
y
x
xy
and
y
x
i
xyi
y
x
i
y
xyi
x
i
yi
x
i
z
and
yi
x
z
Let
47. 47
Unit Three
Solutions/Answers to Exercises 3.1 of page 75
1. Let R be a relation on the set
6
,
5
,
4
,
3
,
2
,
1
A defined by
9
:
,
b
a
b
a
R .
i) List the elements of R
3
,
6
,
2
,
6
,
1
,
6
,
4
,
5
,
3
,
5
,
2
,
5
,
1
,
5
,
5
,
4
,
4
,
4
,
3
,
4
,
2
,
4
,
1
,
4
6
,
3
5
,
3
,
4
,
3
,
3
,
3
,
2
,
3
,
1
,
3
,
6
,
2
,
5
,
2
,
4
,
2
,
3
,
2
,
2
,
2
,
1
,
2
,
6
,
1
5
,
1
,
4
,
1
,
3
,
1
,
2
,
1
,
1
,
1
R
ii) Is 1
R
R Yes since addition is commutative
2. Let R be a relation on the set
7
,
6
,
5
,
4
,
3
,
2
,
1
A defined by
b
a
divides
b
a
R
4
:
,
i) List the elements of R
3
,
7
,
2
,
6
,
1
,
5
R
ii) Find Dom(R) & Range(R)
Dom(R)=
7
,
6
,
5 and Range(R)=
3
,
2
,
1
iii) Find the elements of 1
R
7
,
3
,
6
,
2
,
5
,
1
1
R
iv) Find Dom( 1
R ) & Range( 1
R )
Dom( 1
R )=
3
,
2
,
1 and Range( 1
R )=
7
,
6
,
5
3. Let
6
,
5
,
4
,
3
,
2
,
1
A . Define a relation on A by
1
:
,
x
y
y
x
R . Write down the domain, codomain
and range of R . Find 1
R
Dom( R )
5
,
4
,
3
,
2
,
1
Codomain
6
,
5
,
4
,
3
,
2
,
1
range of R
6
,
5
,
4
,
3
,
2
,
1
1
R =
5
,
6
,
4
,
5
,
3
,
4
,
2
,
3
,
1
,
2
4. Find the domain and range of the relation
.
2
:
,
y
x
y
x
Let G =
.
2
:
,
y
x
y
x
Then Dom(G) = .
: number
real
a
is
x
x
Range(G) = .
: number
real
a
is
y
y
5. Let
8
,
6
,
5
,
3
3
,
2
,
1
B
and . Which of the following are functions from A to B?
a)
3
,
3
,
3
,
2
,
3
,
1
f Function Dom(f) = A and Range(f) =
3
b)
6
,
1
,
5
,
2
,
3
,
1
f Not a function since Dom(f) ≠ A
c)
5
,
2
,
8
,
1
f Not a function since Dom(f) ≠ A
d)
3
,
3
,
5
,
2
,
6
,
1
f Function Dom(f) = A and Range(f) =
6
,
5
,
3
e)
6. Determine the domain and range of the given relation. Is the relation a function?
a)
0
,
2
,
6
,
4
,
5
,
2
,
3
,
4
Domain =
2
,
4
,
2
,
4
Range =
0
,
6
,
5
,
3
It is not a function since the element 2 in the domain maps to more than one element (to 2 different elements:
to -5 and to 0) of the range
48. 48
b)
5
,
1
,
2
3
,
6
,
2
,
8
Domain =
1
,
6
,
8 Range =
5
,
2
3
,
2
It is a function since no element of the domain maps to more than one element of the range
c)
3
,
3
,
1
,
1
,
0
,
0
,
1
,
1
,
3
,
3
Domain =
3
,
1
,
0
,
1
,
3
Range =
0
,
1
,
3
It is a function since no element of the domain maps to more than one element of the range
d)
3
,
3
,
1
,
1
,
8
1
,
3
1
,
1
,
1
,
6
1
,
2
1
Domain =
3
,
1
,
3
1
,
1
,
2
1
Range =
3
,
1
,
8
1
,
1
,
6
1
It is a function since no element of the domain maps to more than one element of the range
e)
5
,
5
,
5
,
4
,
5
,
3
,
5
,
2
,
5
,
1
,
5
,
0
Domain =
5
,
4
,
3
,
2
,
1
,
0 Range =
5
It is a function since no element of the domain maps to more than one element of the range
f)
5
,
5
,
4
,
5
,
3
,
5
,
2
,
5
,
1
,
5
,
0
,
5
Domain =
5 Range =
5
,
4
,
3
,
2
,
1
,
0
It is not a function since the element 5 in the domain maps to more than one element (to 6 different
elements: to 0, to 1, to 2, to 3, to 4 and to 5) of the range
7. Find the domain and the range of the following functions.
a) 2
2
8
1
)
( x
x
x
f
Dom(f) is the set of all real numbers
b)
6
5
1
)
( 2
x
x
x
f
3
2
:
)
(
x
and
x
x
f
Dom
,
3
3
,
2
2
,
c) 8
6
)
( 2
x
x
x
f
,
4
2
,
4
2
:
)
( x
or
x
x
f
Dom
d)
5
2
,
1
2
1
,
4
3
)
(
x
x
x
x
x
f
5
,
1
5
1
:
)
(
x
x
f
Dom
8. Given
1
,
1
1
,
5
3
)
( 2
x
x
x
x
x
f
Find a) f(-3) b) f(1) c) f(6)
a) f(-3) = 3x-5 = 3(-3)-5 = -9-5 = -14
b) f(1) = 0
1
1
1
1
1 2
2
x
c) f(6) = 35
1
36
1
6
1 2
2
x
49. 49
Solutions/Answers to Exercises 3.2 of page 81
1. :
,
3
2
)
( 2
value
each
find
x
x
g
and
x
x
x
f
For
a) )
2
(
g
f b) )
1
(
g
f c) )
3
(
2
g d) )
1
(
fog e) )
1
(
gof f) )
3
(
gog
a)
5
28
5
2
30
5
2
6
3
2
2
2
2
2
2
)
2
( 2
g
f
g
f
b)
4
2
2
4
2
2
3
1
2
1
1
1
1
)
1
(
2
g
f
g
f
c)
9
1
3
1
6
2
3
3
2
3
)
3
(
2
2
2
2
2
g
g
d)
4
3
4
2
1
2
1
4
1
2
1
2
1
2
1
3
1
2
1
)
1
(
2
f
f
g
f
fog
e)
5
2
3
2
2
2
1
1
1
)
1
( 2
g
g
f
g
gof
5
3
10
3
2
3
9
1
2
3
3
1
2
3
1
6
2
3
3
2
3
)
3
(
)
g
g
g
g
g
gog
f
2. If domian
its
state
and
following
the
of
each
for
formula
a
find
x
x
g
and
x
x
f ,
1
2
2
2
a)
x
g
f b)
x
fog c)
x
f
g
d)
x
gof
a)
1
2
2
1
2
1
2
1
2
2
2
3
2
2
x
x
x
x
x
x
x
x
x
x
g
x
f
x
g
f
b)
2
2
2
2
2
2
2
2
2
2
1
6
4
2
1
2
4
2
4
1
1
2
2
4
1
1
2
4
2
1
4
2
1
2
1
2
x
x
x
x
x
x
x
x
x
x
x
x
x
x
f
x
g
f
x
fog
c)
2
2
2
2
1
1
2
2
1
2
2
1
2
2
3
2
2
2
x
x
x
x
x
x
x
x
x
x
f
x
g
x
f
g
d)
1
2
1
2
2
2 2
2
2
x
x
x
g
x
f
g
x
gof
3. x
x
g
x
x
f
Let
2
a) Find domain
its
and
x
fog b) Find domain
its
and
x
gof
c)
Explain
functions
same
the
x
gof
and
and
x
fog
Are ?
numbers
real
all
of
set
the
is
domain
its
and
x
x
x
f
x
g
f
x
fog
a
2
)
njumbers
real
of
set
the
is
domain
its
and
x
x
x
g
x
f
g
x
gof
b
2
2
)
different
are
which
x
x
f
g
and
x
x
g
f
above
b
and
a
see
we
if
example
For
functions
same
the
NOT
are
x
gof
and
x
fog
c
)
)
)
50. 50
4.
7
2
.
3
5
x
x
g
f
that
so
x
g
Fimd
x
x
f
Let
5
10
2
10
2
5
3
7
2
5
7
2
3
5
7
2
x
x
g
x
x
g
x
x
g
x
x
g
x
x
g
f
5.
1
3
.
1
2
x
x
g
f
that
so
x
g
Fimd
x
x
f
Let
2
3
3
2
1
3
1
2
1
3
x
x
g
x
x
g
x
x
g
x
x
g
f
6.
3
1
3
2
.
1
1
x
f
x
f
x
f
that
Show
x
x
x
f
by
defined
function
real
is
f
If
2
2
2
2
3
1
3
2
,
2
2
2
2
2
2
2
1
1
2
2
2
1
2
4
1
2
4
1
2
4
1
2
4
1
3
3
1
1
1
3
3
1
1
3
1
1
1
3
3
3
1
1
1
1
1
3
3
1
3
1
2
1
2
2
x
x
x
f
x
f
x
f
Therefore
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
f
x
f
and
x
x
x
f
and
7. Find tow functions f and g so that the given function
x
fog
x
h where :
a) 3
3
x
x
h b) 3
5
x
x
h c) 1
1
x
x
h d)
6
1
x
x
h
3
3
3
3
.
3
)
x
x
f
x
g
f
x
fog
x
h
Then
x
x
f
and
x
x
g
Let
a
3
5
3
5
.
3
5
)
x
x
f
x
g
f
x
fog
x
h
Then
x
x
f
and
x
x
g
Let
b
1
1
1
.
1
1
)
x
x
f
x
g
f
x
fog
x
h
Then
x
x
f
and
x
x
g
Let
c
6
1
6
.
1
6
)
x
x
f
x
g
f
x
fog
x
h
Then
x
x
f
and
x
x
g
Let
d
8. :
.
1
,
3
4 2
Find
x
x
x
h
and
x
x
g
x
x
f
Let
7
5
)
x
f
a 7
5
)
x
f
b
3
) h
g
f
c
3
.
2
.
1
) h
g
f
d
a
x
f
e
) a
x
f
f
)
25
20
3
28
20
3
7
5
4
7
5
)
x
x
x
x
f
a
8
20
7
15
20
7
3
4
5
7
5
)
x
x
x
x
f
b
x
x
x
x
x
x
x
x
x
x
x
x
x
x
f
x
x
g
f
h
g
f
c
2
2
2
2
2
2
2
2 3
3
4
3
4
3
4
3
1
4
1
3
)
3
2
6
6
2
1
1
3
3
2
1
3
1
4
3
.
2
.
1
) 3
h
g
f
d
3
4
4
3
4
)
a
x
a
x
a
x
f
e
3
4
3
4
)
a
x
a
x
a
x
f
f
51. 51
Solutions/Answers to Exercises 3.3 of page 85
onto
it
Is
one
to
one
f
Is
Z
o
S
from
S
x
x
x
f
function
the
Consider
?
.
int
3
,
2
,
1
,
0
,
1
,
2
,
3
:
,
.
1 2
onto
not
is
f
that
so
Z
f
Range
one
to
one
not
is
f
hence
x
x
x
x
S
x
x
Let
x
x
f
form
the
of
is
function
The
9
,
4
,
1
,
0
, 2
1
2
2
2
1
2
1
2
A
onto
A
from
functions
one
to
one
all
List
A
Let }.
3
,
2
,
1
{
.
2
2
,
3
1
,
2
,
3
,
1
,
1
,
3
2
,
2
,
3
,
1
,
1
,
3
3
,
2
,
2
,
1
,
3
,
3
1
,
2
,
2
,
1
,
2
,
3
3
,
2
,
1
,
1
,
3
,
3
2
,
2
,
1
,
1
6
5
4
3
2
1
f
f
f
f
f
f
y
x
f
A
B
x
y
f
e
i
relation
inverse
the
be
f
Let
B
A
f
Let
:
,
.,
.
,
.
:
.
3 *
*
.
2
2
,
2
,
1
,
2
2
,
2
,
2
,
1
)
*
*
*
function
a
NOT
is
f
means
which
images
different
two
to
maps
where
f
then
f
Let
function
a
be
not
need
f
that
example
an
by
Show
a
function
a
is
f
is
f
means
whic
x
f
y
f
y
y
e
i
x
x
y
y
means
which
x
x
y
f
f
y
f
f
y
y
means
that
x
x
y
y
means
which
x
x
x
f
x
f
and
f
Ramge
f
Dom
Then
is
f
pose
Again
is
f
means
which
x
x
x
f
x
f
x
x
y
y
x
x
y
f
y
f
y
y
and
f
range
f
Dom
Then
x
y
f
or
y
x
f
x
y
f
or
y
x
f
let
and
A
to
in
f
range
from
function
a
is
f
Supose
is
f
if
only
and
if
A
to
in
f
range
from
function
a
is
f
that
Show
b
*
*
2
*
1
*
2
1
2
1
2
1
2
1
2
*
1
*
2
1
2
1
2
1
2
1
2
1
*
2
1
2
1
2
1
2
1
1
2
*
1
*
2
1
*
2
2
*
12
2
1
1
*
1
1
*
*
.,
.
1
1
sup
,
1
1
.
,
1
1
)
done
are
we
above
b
by
Therefore
A
to
in
B
f
range
from
function
a
is
f
Then
B
f
Range
onto
is
f
Since
onto
and
is
f
if
only
and
if
A
to
in
B
from
function
a
is
f
that
Show
c
)
,
1
1
)
*
*
*
1
*
*
*
1
*
.,
.
,
,
.
)
1
1
,
,
)
f
f
e
i
f
of
inverse
an
is
f
Therefore
above
c
by
onto
and
is
f
Therefore
A
to
in
B
from
function
a
is
f
f
f
then
A
to
in
B
from
function
a
is
f
if
that
Show
d
.
B
onto
A
from
function
a
is
x
x
f
by
defined
B
A
f
that
Show
x
R
x
B
and
x
R
x
A
Let
1
1
5
8
5
:
}.
8
5
:
{
}
1
0
:
{
.
4
B
onto
A
from
function
a
is
f
x
x
x
x
x
x
x
f
x
f
that
such
A
x
x
Let
1
1
3
3
5
8
5
5
8
5
, 2
1
2
1
2
1
2
1
2
1
52. 52
5. Which of the following functions are one to one?
R
x
x
f
by
defined
R
R
f
a
,
4
:
)
1
1
4
,
1
1
,
.
&
4
4
,
NOT
but
onto
is
it
means
which
range
the
of
element
one
to
maps
domian
the
of
element
every
words
other
In
NOT
is
f
Therefore
y
x
about
nothing
say
can
we
that
y
f
x
f
that
such
R
y
x
Let
R
x
x
x
f
by
defined
R
R
f
b
,
1
6
:
)
1
1
6
6
1
6
1
6
,
is
f
y
x
y
x
y
x
y
f
x
f
that
such
R
y
x
Let
R
x
x
x
f
by
defined
R
R
f
c
,
7
:
) 2
3
3
16
7
9
7
3
7
3
3
3
1
1
7
7
,
2
2
2
2
2
2
but
f
f
example
counter
a
As
NOT
is
f
means
which
y
x
y
x
y
x
y
f
x
f
that
such
R
y
x
Let
R
x
x
x
f
by
defined
R
R
f
d
,
:
) 3
1
1
, 3
3
is
f
y
x
y
x
y
f
x
f
that
such
R
y
x
Let
}
7
{
,
7
1
2
}
7
{
:
) R
x
x
x
x
f
by
defined
R
R
f
e
1
1
15
15
7
7
14
14
2
2
7
14
2
7
14
2
1
2
7
1
2
7
7
1
2
7
1
2
}
7
{
,
is
f
y
x
y
x
y
y
x
x
xy
xy
y
x
xy
y
x
xy
y
x
x
y
y
y
x
x
y
f
x
f
that
such
R
y
x
Let
6.Which of the following functions are onto?
onto
is
f
therefore
R
f
Range
R
x
x
x
f
by
defined
R
R
f
a
,
,
,
49
115
:
)
onto
NOT
is
f
therefore
R
f
Range
R
x
x
x
f
by
defined
R
R
f
b
,
,
,
0
,
:
)
onto
NOT
is
f
therefore
R
f
Range
R
x
x
x
f
by
defined
R
R
f
c
,
,
,
0
,
:
) 2
onto
NOT
is
f
therefore
R
f
Range
R
x
x
x
f
by
defined
R
R
f
d
,
,
,
4
,
4
:
) 2
53. 53
7. if
x
f
Find 1
7
6
7
6
7
6
6
7
6
7
)
1
1
x
x
f
y
x
y
x
y
x
y
f
x
x
f
a
2
9
4
2
9
4
2
9
4
9
2
4
4
9
2
4
9
2
)
1
1
x
x
f
x
y
y
x
y
x
y
x
y
f
x
x
f
b
x
x
f
y
x
y
x
y
x
y
x
y
f
x
x
f
c
1
1
1
1
1
1
1
1
1
1
1
1
)
1
1
1
3
4
1
3
4
3
4
1
3
3
4
3
1
3
3
3
4
3
1
3
3
4
3
1
3
1
3
4
3
3
4
3
4
3
4
)
1
1
x
x
f
y
x
y
x
y
x
y
x
y
x
y
y
y
y
y
y
x
y
f
x
x
x
f
d
x
x
x
f
x
x
y
x
x
y
x
xy
y
x
xy
y
y
xy
x
y
y
x
y
y
x
y
f
x
x
x
f
e
2
5
3
2
5
3
3
2
5
3
2
5
3
2
5
3
5
2
3
5
2
1
2
1
3
5
2
1
3
5
)
1
1
1
1
1
1
1
)
3
1
3
3
3
1
3
x
x
f
y
x
y
x
y
x
y
f
x
x
f
f
1
2
1
2
2
1
2
1
2
1
1
2
1
2
)
1
2
2
2
1
2
x
x
f
x
y
y
x
y
x
y
x
y
x
y
f
x
x
f
g
x
x
x
f
y
x
x
x
y
x
y
xy
x
y
y
x
y
y
x
y
f
x
x
x
f
h
2
2
2
2
2
1
1
2
1
2
)
1
1
54. 54
Solutions/Answers to Exercises 3.4 of page 97 – 98
1. Perform the requested divisions. Find the quotient and the remainder and verify the Remainder Theorem
by computing p(a).
4
8
5
) 2
x
by
x
x
x
p
Divide
a
9
44
36
9
8
9
4
8
5
4
2
2
x
x
x
x
x
x
x
x Therefore, the quotient is x-9 and the remainder is 44
Theorem
mainder
Verified
p Re
44
8
20
16
8
4
5
4
4
2
Therefore, the quotient is 1
2
3
x
x
x
and no remainder (the remainder is 0)
Theorem
mainder
the
by
Verified
p Re
0
1
1
1 4
Theorem
mainder
by
Verified
p
is
remainder
the
x
x
is
quotient
the
Therefore
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
by
x
x
x
x
p
Divide
b
Re
24
8
112
128
4
4
4
7
4
2
4
24
5
2
,
5
2
24
20
5
4
5
4
4
8
2
4
7
2
4
4
4
7
2
)
2
3
2
2
2
2
2
3
2
3
2
3
1
0
1
1
1
1
1
1
1
1
)
2
3
2
2
2
3
3
4
3
4
4
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
by
x
x
p
Divide
c
55. 55
6
3
3
is
quotient
the
Therefore,
3
3
6
3
3
3
3
3
3
3
3
3
2
3
2
3
2
1
1
3
2
)
2
3
4
2
3
4
2
2
2
3
2
3
3
4
2
4
4
5
2
5
2
5
is
remainder
the
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
by
x
x
x
p
Divide
d
Theorem
mainder
the
by
Verified
p Re
6
3
2
1
3
1
2
1
1
2
5
2
3
2
0
8
2
8
2
12
3
8
10
3
8
2
8
10
11
2
4
4
0
4
.
8
10
11
2
,
0
4
.
2
2
2
2
2
3
2
3
2
3
x
x
x
x
x
x
x
x
x
x
x
x
x
x
p
of
factor
is
x
p
possible
as
completely
as
x
x
x
x
p
factor
p
that
Given
4
1
2
2
4
2
3
2
8
10
11
2
, 2
2
3
x
x
x
x
x
x
x
x
x
Therefore
36
4
0
9
36
9
36
4
9
36
4
4
1
1
4
4
1
0
4
1
.
0
4
1
,
9
36
4
.
3
2
2
3
2
3
2
3
x
x
x
x
x
x
x
x
x
r
of
factor
a
is
x
or
x
r
x
ofr
zeros
remaining
the
find
r
x
x
x
x
r
that
Given
r
of
roots
or
zeros
the
are
x
x
x
Therefore
x
x
x
x
x
x
x
x
x
x
x
r
Now
3
,
3
,
4
1
,
3
3
1
4
4
9
1
4
4
36
4
1
4
9
36
4 2
2
2
3
56. 56
10
3
0
90
60
10
90
60
10
27
18
3
90
87
28
3
9
6
90
87
19
3
9
6
9
6
3
3
.
.
90
87
19
3
3
.
4
2
2
2
2
3
2
3
2
3
4
2
3
4
2
2
2
2
3
4
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
p
of
factor
a
is
x
x
or
x
p
of
zero
double
a
x
p
of
zeros
the
all
Find
x
x
x
x
x
p
of
zero
double
a
that
Given
p
f
zeros
the
are
x
and
x
x
Hence
x
x
x
x
x
x
x
x
x
x
x
p
Therefore
2
5
,
3
,
2
5
3
10
3
3
90
87
19
3
,
2
2
2
2
3
4
5. a) Write the general polynomial p(x) whose only zeros are 1, 2, and 3, with multiplicity 3, 2 and 1 respectively.
What is its degree?
3)
-
(x
2)
-
(x
1)
-
k(x
=
p(x) 2
3
b) Find p(x) described in part (a) if p(0) = 6
6
deg
3)
-
(x
2)
-
(x
1)
-
(x
2
1
,
2
1
k
6
k
12
6
3)
k(-1)(4)(-
6
(-3)
(-2)
k(-1)
6
3)
-
(0
2)
-
(0
1)
-
k(0
6
p(o)
and
3)
-
(x
2)
-
(x
1)
-
k(x
=
p(x)
2
3
2
3
2
3
2
3
is
ree
the
And
x
p
Therefore
57. 57
i
x
i
x
x
i
x
i
x
i
x
i
x
x
i
x
i
x
9
6
6
1
2
0
39
9
6
39
9
6
9
20
6
1
39
14
6
1
6
4
2
39
14x
5x
-
2x
3i
-
2
-
x
x
p
of
zeros
remainig
the
,
39
14x
5x
-
2x
=
p(x)
of
root
a
is
3i
-
2
If
6.
2
2
2
2
3
2
3
2
3
x
p
of
roots
other
the
are
i
i
x
or
i
i
x
i
i
i
i
i
x
i
i
i
i
i
i
x
a
ac
b
b
x
formula
quadiratic
the
use
we
i
x
i
x
factorize
To
4
13
84
6
1
4
13
84
6
1
4
13
84
6
1
4
35
48
72
12
6
1
4
72
48
35
12
6
1
2
2
9
6
2
4
6
1
6
1
2
4
:
9
6
6
1
2
2
2
2
7. Determine the rational zeros of the polynomials:
10
3
0
10
10
10
10
3
3
10
7
3
10
7
4
1
1
0
10
7
4
1
10
1
7
1
4
1
:
1
10
7
4
)
2
2
2
2
3
2
3
2
3
2
3
x
x
x
x
x
x
x
x
x
x
x
x
x
x
p
of
factor
one
is
x
x
x
x
x
x
p
a
2
,
5
2
7
3
2
40
9
3
2
10
1
4
3
3
:
,
10
3
,
10
3
1
10
7
4
,
2
2
2
2
3
x
x
x
formula
quadratic
use
we
x
x
factorize
to
Again
x
x
x
x
x
x
x
p
Therefore
58. 58
2
5
,
1
2
5
1
10
7
4 2
3
x
and
x
x
are
zeros
rational
the
that
So
x
x
x
x
x
x
x
p
5
11
2
0
15
5
15
5
33
11
15
28
11
6
2
15
28
5
2
3
.
3
0
99
99
15
84
45
54
15
3
28
3
5
3
2
:
3
15
28
5
2
)
2
2
2
2
3
2
3
2
3
2
3
x
x
x
x
x
x
x
x
x
x
x
x
x
x
p
of
root
one
is
x
means
which
x
x
x
x
x
p
b
zeros
l
therationa
are
x
x
x
then
and
x
x
x
x
x
x
Therefore
x
or
x
x
formula
quadratic
g
u
again
x
x
factorize
to
need
we
therefore
and
x
x
x
x
x
x
Hence
2
1
,
5
,
3
1
2
5
3
15
28
5
2
,
2
1
,
5
4
9
11
4
81
11
4
40
121
11
4
5
2
4
11
11
:
sin
5
11
2
5
11
2
3
15
28
5
2
,
2
3
2
2
2
2
3
1
4
6
) 2
3
x
x
x
x
p
c
1
5
6
0
1
1
5
5
1
4
5
6
6
1
4
6
1
1
0
1
4
1
6
1
1
4
1
1
6
:
1
2
2
2
2
3
2
3
2
3
x
x
x
x
x
x
x
x
x
x
x
x
x
x
p
of
root
one
is
x
means
which
x
59. 59
zeros
l
therationa
are
x
x
x
then
and
x
x
x
x
x
x
Therefore
x
or
x
x
formula
quadratic
g
u
again
x
x
factorize
to
need
we
therefore
and
x
x
x
x
x
x
Hence
3
1
,
2
1
,
1
1
3
1
2
1
1
4
6
,
3
1
12
4
,
2
1
12
6
12
1
5
12
1
5
12
24
25
5
12
1
6
4
5
5
:
sin
1
5
6
1
5
6
1
1
4
6
,
2
3
2
2
2
2
3
8. Find the domain and the real zeros of the given function.
a)
25
3
2
x
x
f
5
,
5
5
,
5
0
25
: 2
R
nimbers
real
all
x
x
f
Dom
5
25
25
0
25
: 2
2
x
x
x
x
f
Dom
Real zeros: .
3
25
0
0
25
3
0 2
2
zero
real
no
is
there
that
so
false
is
which
x
x
x
f
b) b)
12
4
12
4
12
4
3 2
2
2
x
x
as
x
x
Correcting
x
x
x
x
g
0
12
4
: 2
x
x
x
f
Dom
6
2
2
8
4
2
64
4
2
48
16
4
2
12
4
16
4
0
12
4
: 2
x
or
x
x
x
x
x
x
x
f
Dom
6
,
2
6
,
2
:
R
x
x
x
f
Dom
Real zero: .
3
3
0
3 zero
real
the
is
means
which
x
x
or
12
4
12
4
12
4
3 2
2
2
x
x
as
x
x
Correcting
x
x
x
x
g
6
2
2
8
4
2
64
4
2
48
16
4
2
12
4
16
4
0
12
4
: 2
x
or
x
x
x
x
x
x
x
f
Dom
6
,
2
6
,
2
:
R
x
x
x
f
Dom
Real zero: .
3
3
0
3 zero
real
the
is
means
which
x
x
c)
x
x
x
x
x
f
2
3
3
2
3
2
0
2
3
: 2
3
x
x
x
x
f
Dom