This document discusses mathematical language and symbols. It covers topics like characteristics of mathematical language, expressions versus sentences, conventions in mathematical language, and four basic concepts including the language of sets. Key points include that mathematical language uses both natural language terms and specialized symbolic notation for formulas. Expressions represent objects but do not form a complete thought, while sentences make a statement that can be true or false. Conventions like PEMDAS and specific mathematical terms are agreed upon. Set theory concepts explained include finite and infinite sets, operations on sets like union and intersection, and diagrams like Venn diagrams.
The document discusses the language and conventions of mathematics. It outlines topics that will be covered, including characteristics of mathematical language, expressions versus sentences, conventions in mathematical language, four basic concepts, elementary logic, and formality. Several sections provide examples and definitions of key concepts in mathematical language and sets, such as expressions, sentences, finite and infinite sets, unions, intersections, complements, and operations on sets. The overall document serves as an introduction to mathematical language and sets.
This document discusses geometric sequences and provides examples. It defines a geometric sequence as a sequence where each term is found by multiplying or dividing the same value from one term to the next. An example given is 2, 4, 8, 16, 32, 64, 128, etc. The document also provides the general formula for a geometric sequence as {a, ar, ar2, ar3, ...} where a is the first term and r is the common ratio between terms. It gives practice problems for finding missing terms and the common ratio of geometric sequences.
This action research study examined the effects of activity-based teaching methods on 7th grade students' understanding of adding and subtracting integers. The study involved 26 students who completed pre- and post-tests on integer addition. Between the tests, students learned about integers using group work, interviews, math lab activities, notebooks, games, and debates. Results showed students significantly improved their conceptual understanding and procedural skills, with the average test score increasing from 63.85 to 90.77. Specifically, students improved most at adding negative integers and adding negative and positive integers. The study concluded activity-based learning is effective for teaching integers and benefits students' mathematics performance.
Math 4 axioms on the set of real numbersLeo Crisologo
The document discusses the axioms that define a field and the set of real numbers. It outlines the closure, associativity, commutativity, distributive property, identity element, and inverse element axioms for fields. It also covers equality axioms and proves theorems like cancellation for addition and the involution property using the field axioms.
A matrix is a rectangular array of numbers arranged in rows and columns. The dimensions of a matrix are written as the number of rows x the number of columns. Each individual entry in the matrix is named by its position, using the matrix name and row and column numbers. Matrices can represent systems of equations or points in a plane. Operations on matrices include addition, multiplication by scalars, and dilation of points represented by matrices.
Factor Theorem and Remainder Theorem. Mathematics10 Project under Mrs. Marissa De Ocampo. Prepared by Danielle Diva, Ronalie Mejos, Rafael Vallejos and Mark Lenon Dacir of 10- Einstein. CNSTHS.
Angles Formed by Parallel Lines Cut by a TransversalBella Jao
This document discusses parallel lines and angles formed when lines are cut by a transversal. It begins by defining parallel lines and explaining that a transversal is a line that intersects two or more lines at different points. It then defines and provides examples of several types of angles formed, including alternate interior/exterior angles, same side interior/exterior angles, and corresponding angles. Groups are then assigned to prove properties of these angles for parallel lines cut by a transversal. The document concludes with practice problems finding missing angle measures using the properties and an assignment involving parallel lines cut by a transversal.
The document discusses the language and conventions of mathematics. It outlines topics that will be covered, including characteristics of mathematical language, expressions versus sentences, conventions in mathematical language, four basic concepts, elementary logic, and formality. Several sections provide examples and definitions of key concepts in mathematical language and sets, such as expressions, sentences, finite and infinite sets, unions, intersections, complements, and operations on sets. The overall document serves as an introduction to mathematical language and sets.
This document discusses geometric sequences and provides examples. It defines a geometric sequence as a sequence where each term is found by multiplying or dividing the same value from one term to the next. An example given is 2, 4, 8, 16, 32, 64, 128, etc. The document also provides the general formula for a geometric sequence as {a, ar, ar2, ar3, ...} where a is the first term and r is the common ratio between terms. It gives practice problems for finding missing terms and the common ratio of geometric sequences.
This action research study examined the effects of activity-based teaching methods on 7th grade students' understanding of adding and subtracting integers. The study involved 26 students who completed pre- and post-tests on integer addition. Between the tests, students learned about integers using group work, interviews, math lab activities, notebooks, games, and debates. Results showed students significantly improved their conceptual understanding and procedural skills, with the average test score increasing from 63.85 to 90.77. Specifically, students improved most at adding negative integers and adding negative and positive integers. The study concluded activity-based learning is effective for teaching integers and benefits students' mathematics performance.
Math 4 axioms on the set of real numbersLeo Crisologo
The document discusses the axioms that define a field and the set of real numbers. It outlines the closure, associativity, commutativity, distributive property, identity element, and inverse element axioms for fields. It also covers equality axioms and proves theorems like cancellation for addition and the involution property using the field axioms.
A matrix is a rectangular array of numbers arranged in rows and columns. The dimensions of a matrix are written as the number of rows x the number of columns. Each individual entry in the matrix is named by its position, using the matrix name and row and column numbers. Matrices can represent systems of equations or points in a plane. Operations on matrices include addition, multiplication by scalars, and dilation of points represented by matrices.
Factor Theorem and Remainder Theorem. Mathematics10 Project under Mrs. Marissa De Ocampo. Prepared by Danielle Diva, Ronalie Mejos, Rafael Vallejos and Mark Lenon Dacir of 10- Einstein. CNSTHS.
Angles Formed by Parallel Lines Cut by a TransversalBella Jao
This document discusses parallel lines and angles formed when lines are cut by a transversal. It begins by defining parallel lines and explaining that a transversal is a line that intersects two or more lines at different points. It then defines and provides examples of several types of angles formed, including alternate interior/exterior angles, same side interior/exterior angles, and corresponding angles. Groups are then assigned to prove properties of these angles for parallel lines cut by a transversal. The document concludes with practice problems finding missing angle measures using the properties and an assignment involving parallel lines cut by a transversal.
Slope and y intercept in real world examplesJessica Polk
This document provides examples of how to interpret slope and y-intercept in real-world contexts. It defines slope as the rate of change and y-intercept as the value of y when x is zero. Examples given include using slope to show the growth rate of a tree over time, and using the y-intercept to represent an initial debt amount before payments are made. It also works through an example of using slope-intercept form to model the melting of an ice cream cone over time and calculate how tall it would be after 3 minutes.
This document discusses integers and absolute value. It defines integers as positive and negative whole numbers and explains how to graph and order integers on a number line. It also defines absolute value as the distance from zero and how to evaluate the absolute value of integers, including that the absolute value of a number can never be negative. Examples are provided for graphing, ordering, and finding opposites and absolute values of integers.
The document provides information about triangle congruence, including:
1. There are three postulates for proving triangles are congruent: side-side-side (SSS), side-angle-side (SAS), and angle-side-angle (ASA).
2. The SSS postulate states that if three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent.
3. The SAS postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent.
4. The ASA postulate states that if two angles
This document contains definitions and examples of postulates and theorems in geometry. It defines a postulate as a statement accepted as true without proof, while a theorem is an important statement that must be proved. It lists several postulates, including that a line contains at least two points, through any two points there is exactly one line, and through any three noncollinear points there is exactly one plane. It also lists some theorems, such as if two lines intersect then they intersect at exactly one point, and if two lines intersect then exactly one plane contains the lines.
The document discusses sets and Venn diagrams. It provides examples of sets representing students' favorite subjects and animals that can be categorized as water animals or two-legged animals. It explains that the intersection of sets contains elements that are common to both sets, while the union of sets contains all unique elements of both sets. An example Venn diagram shows the intersection and union of sets A and B. Questions are provided about representing sets using Venn diagrams and calculating unions and intersections of given sets.
The document discusses properties and laws of exponents, radicals, logarithms including the definition of rational exponents, properties of logarithms such as the change of base formula, and examples of simplifying expressions using exponent laws, combining like radicals, and solving logarithmic equations by using properties of logarithms. It also provides sample problems and their step-by-step solutions for simplifying expressions and solving equations involving exponents, radicals, and logarithms.
the rectangular coordinate system and midpoint formulas, linear equations in two variable, slope of a line, equation of a line, applications of linear equations and graphing
Lesson plan on Linear inequalities in two variablesLorie Jane Letada
This document contains a semi-detailed lesson plan for a math class on linear inequalities in two variables. The lesson plan outlines intended learning outcomes, learning content including subject matter and reference materials, learning experiences including sample math word problems and explanations of key concepts, an evaluation through an online quiz, and an assignment for students to create a budget proposal applying their understanding of linear inequalities.
The document discusses relations and functions in mathematics. It provides an overview of key concepts to be covered, including set-builder notation, the rectangular coordinate system, and different ways to represent relations and functions using tables, mappings, graphs and rules. The objectives are for students to understand these concepts and be able to identify, illustrate, and determine different types of relations and functions.
This document discusses integers and absolute values. It defines integers as numbers to the left and right of zero, with negative integers being less than zero and positive integers being greater than zero. Absolute value is defined as the distance a number is from zero on the number line. Examples demonstrate graphing integers on a number line and performing calculations with absolute values.
This document defines basic set theory concepts including sets, elements, notation for sets, subsets, unions, intersections, and empty sets. It provides examples and notation for these concepts. Sets are collections of objects, represented with capital letters, while elements are individual objects represented with lowercase letters. Notation includes braces { } to list elements and vertical bars for set builder notation. A is a subset of B if all elements of A are also in B, written as A ⊆ B. The union of sets A and B contains all elements that are in A or B, written as A ∪ B. The intersection of sets A and B contains elements that are only in both A and B, written as A ∩ B.
Linear Equations and Inequalities in One Variablemisey_margarette
The document discusses linear equations and inequalities in one variable. It defines linear equations and inequalities, and describes methods for solving them including: guess-and-check, cover-up, and working backwards. It also covers properties of equality and inequality, and provides examples of solving linear equations and inequalities using these properties and graphical representations of solution sets on number lines.
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
Multiplying Polynomials: Two BinomialsJoey Valdriz
This document contains notes from a mathematics lesson on multiplying polynomials. The key points covered are:
1. The learner recalls the laws of exponents and multiplies two binomials.
2. Examples are provided of multiplying polynomials using algebra tiles, the distributive property, the box method, and FOIL (First, Outer, Inner, Last).
3. Practice problems are given for students to multiply different binomial expressions.
This document discusses four methods for graphing linear equations on a coordinate plane:
1. Using any two points on the line.
2. Using the x-intercept and y-intercept.
3. Using the slope and y-intercept.
4. Using the slope and one known point.
Examples are provided to illustrate each method. Graphing linear equations is important for visualizing relationships between variables in real-life situations.
This document discusses arithmetic sequences and their properties. It defines an arithmetic sequence as a sequence where the difference between consecutive terms is constant. It provides the formula for the nth term of an arithmetic sequence as an = a1 + (n-1)d, where a1 is the first term, n is the term number, and d is the common difference. It gives examples of finding specific terms and summarizing sequences. It also discusses the arithmetic mean and arithmetic sum formulas.
The document discusses several theorems related to triangles:
1) The triangle inequality theorem states that the length of any side of a triangle must be less than the sum of the lengths of the other two sides and greater than their difference.
2) The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the remote interior angles.
3) The hinge theorem states that if two triangles have two congruent sides but not congruent included angles, then the third sides will be unequal, with the longer side opposite the larger angle.
This document introduces set theory and its importance and applications. It defines what a set is and provides examples of different types of sets such as finite, infinite, equal, subset, power and universal sets. It describes operations on sets like union, intersection and complements. The document discusses the history of set theory and its founder Georg Cantor. It provides examples of how set theory is applied in business organization and security. Venn diagrams are introduced as a way to visualize sets. An example problem is presented to demonstrate applying set theory and Venn diagrams. The document finds that set theory is widely used in many disciplines and can be applied at different levels in business operations for problems involving intersecting and non-intersecting sets.
Slope and y intercept in real world examplesJessica Polk
This document provides examples of how to interpret slope and y-intercept in real-world contexts. It defines slope as the rate of change and y-intercept as the value of y when x is zero. Examples given include using slope to show the growth rate of a tree over time, and using the y-intercept to represent an initial debt amount before payments are made. It also works through an example of using slope-intercept form to model the melting of an ice cream cone over time and calculate how tall it would be after 3 minutes.
This document discusses integers and absolute value. It defines integers as positive and negative whole numbers and explains how to graph and order integers on a number line. It also defines absolute value as the distance from zero and how to evaluate the absolute value of integers, including that the absolute value of a number can never be negative. Examples are provided for graphing, ordering, and finding opposites and absolute values of integers.
The document provides information about triangle congruence, including:
1. There are three postulates for proving triangles are congruent: side-side-side (SSS), side-angle-side (SAS), and angle-side-angle (ASA).
2. The SSS postulate states that if three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent.
3. The SAS postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent.
4. The ASA postulate states that if two angles
This document contains definitions and examples of postulates and theorems in geometry. It defines a postulate as a statement accepted as true without proof, while a theorem is an important statement that must be proved. It lists several postulates, including that a line contains at least two points, through any two points there is exactly one line, and through any three noncollinear points there is exactly one plane. It also lists some theorems, such as if two lines intersect then they intersect at exactly one point, and if two lines intersect then exactly one plane contains the lines.
The document discusses sets and Venn diagrams. It provides examples of sets representing students' favorite subjects and animals that can be categorized as water animals or two-legged animals. It explains that the intersection of sets contains elements that are common to both sets, while the union of sets contains all unique elements of both sets. An example Venn diagram shows the intersection and union of sets A and B. Questions are provided about representing sets using Venn diagrams and calculating unions and intersections of given sets.
The document discusses properties and laws of exponents, radicals, logarithms including the definition of rational exponents, properties of logarithms such as the change of base formula, and examples of simplifying expressions using exponent laws, combining like radicals, and solving logarithmic equations by using properties of logarithms. It also provides sample problems and their step-by-step solutions for simplifying expressions and solving equations involving exponents, radicals, and logarithms.
the rectangular coordinate system and midpoint formulas, linear equations in two variable, slope of a line, equation of a line, applications of linear equations and graphing
Lesson plan on Linear inequalities in two variablesLorie Jane Letada
This document contains a semi-detailed lesson plan for a math class on linear inequalities in two variables. The lesson plan outlines intended learning outcomes, learning content including subject matter and reference materials, learning experiences including sample math word problems and explanations of key concepts, an evaluation through an online quiz, and an assignment for students to create a budget proposal applying their understanding of linear inequalities.
The document discusses relations and functions in mathematics. It provides an overview of key concepts to be covered, including set-builder notation, the rectangular coordinate system, and different ways to represent relations and functions using tables, mappings, graphs and rules. The objectives are for students to understand these concepts and be able to identify, illustrate, and determine different types of relations and functions.
This document discusses integers and absolute values. It defines integers as numbers to the left and right of zero, with negative integers being less than zero and positive integers being greater than zero. Absolute value is defined as the distance a number is from zero on the number line. Examples demonstrate graphing integers on a number line and performing calculations with absolute values.
This document defines basic set theory concepts including sets, elements, notation for sets, subsets, unions, intersections, and empty sets. It provides examples and notation for these concepts. Sets are collections of objects, represented with capital letters, while elements are individual objects represented with lowercase letters. Notation includes braces { } to list elements and vertical bars for set builder notation. A is a subset of B if all elements of A are also in B, written as A ⊆ B. The union of sets A and B contains all elements that are in A or B, written as A ∪ B. The intersection of sets A and B contains elements that are only in both A and B, written as A ∩ B.
Linear Equations and Inequalities in One Variablemisey_margarette
The document discusses linear equations and inequalities in one variable. It defines linear equations and inequalities, and describes methods for solving them including: guess-and-check, cover-up, and working backwards. It also covers properties of equality and inequality, and provides examples of solving linear equations and inequalities using these properties and graphical representations of solution sets on number lines.
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
Multiplying Polynomials: Two BinomialsJoey Valdriz
This document contains notes from a mathematics lesson on multiplying polynomials. The key points covered are:
1. The learner recalls the laws of exponents and multiplies two binomials.
2. Examples are provided of multiplying polynomials using algebra tiles, the distributive property, the box method, and FOIL (First, Outer, Inner, Last).
3. Practice problems are given for students to multiply different binomial expressions.
This document discusses four methods for graphing linear equations on a coordinate plane:
1. Using any two points on the line.
2. Using the x-intercept and y-intercept.
3. Using the slope and y-intercept.
4. Using the slope and one known point.
Examples are provided to illustrate each method. Graphing linear equations is important for visualizing relationships between variables in real-life situations.
This document discusses arithmetic sequences and their properties. It defines an arithmetic sequence as a sequence where the difference between consecutive terms is constant. It provides the formula for the nth term of an arithmetic sequence as an = a1 + (n-1)d, where a1 is the first term, n is the term number, and d is the common difference. It gives examples of finding specific terms and summarizing sequences. It also discusses the arithmetic mean and arithmetic sum formulas.
The document discusses several theorems related to triangles:
1) The triangle inequality theorem states that the length of any side of a triangle must be less than the sum of the lengths of the other two sides and greater than their difference.
2) The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the remote interior angles.
3) The hinge theorem states that if two triangles have two congruent sides but not congruent included angles, then the third sides will be unequal, with the longer side opposite the larger angle.
This document introduces set theory and its importance and applications. It defines what a set is and provides examples of different types of sets such as finite, infinite, equal, subset, power and universal sets. It describes operations on sets like union, intersection and complements. The document discusses the history of set theory and its founder Georg Cantor. It provides examples of how set theory is applied in business organization and security. Venn diagrams are introduced as a way to visualize sets. An example problem is presented to demonstrate applying set theory and Venn diagrams. The document finds that set theory is widely used in many disciplines and can be applied at different levels in business operations for problems involving intersecting and non-intersecting sets.
The document discusses the key differences between the English and mathematics languages, including how words, symbols, and concepts have different meanings or representations. It explains how mathematics has developed a precise symbolic language to concisely express relationships, operations, and concepts in a way that is internationally understood regardless of spoken language. Precise definitions and notations are provided for important mathematical concepts like sets, relations, functions, and equations.
The document discusses the key topics in discrete mathematics that will be covered across 5 units. Unit 1 covers sets, relations, functions and their properties. Unit 2 discusses mathematical induction, counting techniques, and number theory topics. Unit 3 is on propositional logic, logical equivalence and proof techniques. Unit 4 covers algebraic structures like groups, rings and fields. Unit 5 is about graphs, trees, and their properties like coloring and shortest paths. The document also lists 3 recommended textbooks for the course.
Set theory is the branch of mathematics that studies sets, which are collections of objects. A set is well-defined if we can determine whether an object is an element of that set. The empty or null set is represented by the symbol Ø and contains no elements. A set can be either finite, containing a whole number of elements, or infinite.
The document provides an introduction to set theory. It defines what a set is and discusses different ways of representing sets using roster form and set-builder form. It also defines types of sets such as the empty set, singleton set, finite sets, and equivalent sets. Subsets are introduced, including proper and improper subsets. Important subsets of the real numbers like the natural numbers, integers, rational numbers, and irrational numbers are identified. Intervals are also discussed as subsets of the real line.
This document provides an introduction to set theory. It begins with definitions of fundamental set concepts like elements, membership, representation of sets in roster and set-builder forms, empty and singleton sets, finite and infinite sets, equal and equivalent sets. It then discusses types of sets such as subsets and proper subsets, the power set of a set, and universal sets. Examples are provided to illustrate each concept. The document also introduces Venn diagrams to represent relationships between sets.
Business mathematics is a very powerful tools and analytic process that resul...mkrony
Business mathematics is a powerful analytical tool that can result in optimal solutions despite limitations. The document lists the names and IDs of 7 group members working on topics related to permutations, combinations, number systems, set theory, and linear programming. It provides examples and definitions of permutations, combinations, and the differences between them.
This document defines several key concepts in set theory, including:
1) A set is a well-defined collection of distinct objects that can include numbers, letters, objects, etc. Sets are represented using curly brackets and elements are separated by commas.
2) Elements are the individual objects that make up a set. Sets can be defined using a tabular method by listing elements or a set-builder method using properties of the elements.
3) There are different types of sets such as finite sets with a countable number of elements, infinite sets, singleton/unit sets with one element, and the empty set with no elements.
The document discusses variables, sets, relations, functions, and logic. It defines variables as symbols that represent numbers or values. It describes two methods for describing sets - the roster method of listing elements and the set-builder method of describing common characteristics. Relations are defined as sets of ordered pairs, and functions as mappings where each input has a single output. Logic involves propositions that can be true or false and operations on propositions like conjunction assessed with truth tables.
A survey was conducted of 1,000 people who like to travel in Phetchabun. The following information was found:
- 320 people like to travel by bus
- 240 people like to travel by car
- 560 people like to travel by motorcycle
- 100 people like to travel by bus and car
- 50 people like to travel by bus and motorcycle
- 40 people like to travel by car and motorcycle
- 30 people like to travel by bus, car, and motorcycle
The number of people who like to travel by at least one mode is the total of all the categories listed above. Therefore, the number of people who like to travel by at least one way is 320 + 240 + 560 +
This document provides an introduction to the concepts of sets in mathematics. It begins by defining what a set is and how sets are described and classified. Examples are given of different types of sets, including universal sets, unit sets, finite sets, infinite sets, and the empty set. Operations on sets such as unions, intersections, and differences are then explained through examples. The document concludes by relating sets to real-world scenarios and encouraging students not to fear mistakes when trying new things.
This document outlines the topics to be covered in a discrete mathematics course, including sets, logic, proofs, counting, combinatorics, relations, graphs/trees, and Boolean functions. It provides examples of some key concepts in sets like subsets, unions, and intersections. It also introduces basic concepts in logic such as statements, negation, conjunction, disjunction, and implication. Sample truth tables are provided to illustrate logical connectives.
The document discusses metric dimensions of metric spaces. It defines metric dimension as the minimum number of points needed to resolve a metric space. It provides examples of calculating the metric dimension for various spaces, such as classical geometric spaces having dimension n having metric dimension n+1. It also discusses metric dimensions of subspaces, proving there can exist a subspace with higher metric dimension than the original space. It gives an example using points around a circle to illustrate this case.
The document defines set concepts and their properties. It explains that a set is a collection of distinct objects, called elements. Properties discussed include how sets are denoted, elements belong or don't belong to sets, order doesn't matter, and counting each element once. It also covers set theory, Venn diagrams, ways to represent sets, types of sets (empty, finite, infinite), equal sets, subsets, cardinality, and operations like intersection, union, difference, and complement of sets. Examples are provided to illustrate each concept.
This document provides an overview of key concepts in set theory, including:
1) Sets can be represented in roster or set-builder form. Common sets used in mathematics include the natural numbers, integers, rational numbers, and real numbers.
2) The empty set contains no elements. Finite sets have a definite number of elements, while infinite sets have an unlimited number of elements.
3) Two sets are equal if they contain exactly the same elements. A set is a subset of another set if all its elements are also elements of the other set.
4) The power set of a set contains all possible subsets of that set.
This document outlines lecture material on predicate calculus from a discrete mathematics course taught by Dr. D. Ezhilmaran and M. Adhiyaman at VIT University in Tamil Nadu, India. It defines predicates, the universe of discourse, universal and existential quantifiers, and provides examples of translating statements using these logical concepts. The document also presents solutions to several problems applying the rules of predicate calculus and inference.
Chapter 2 Mathematical Language and Symbols.pdfRaRaRamirez
This document discusses mathematical language and symbols. It defines key concepts such as sets, relations, functions, and binary operations. Sets are collections of distinct objects that can be defined using a roster or rule. Relations pair elements between two sets. A function is a special type of relation where each input is paired with exactly one output. Binary operations take two inputs from a set and return an output in that same set. Common properties of binary operations include commutativity and associativity.
The document discusses sets, relations, and functions. It begins by providing examples of well-defined and not well-defined collections to introduce the concept of a set. A set is defined as a collection of well-defined objects. Standard notations for sets are introduced. Sets can be represented using a roster method by listing elements or a set-builder form using a common property. Sets are classified as finite or infinite based on the number of elements, and other set types like the empty set and singleton set are defined. Equal, equivalent, and disjoint sets are also defined.
Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.pptHenry Hollis
The History of NZ 1870-1900.
Making of a Nation.
From the NZ Wars to Liberals,
Richard Seddon, George Grey,
Social Laboratory, New Zealand,
Confiscations, Kotahitanga, Kingitanga, Parliament, Suffrage, Repudiation, Economic Change, Agriculture, Gold Mining, Timber, Flax, Sheep, Dairying,
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إضغ بين إيديكم من أقوى الملازم التي صممتها
ملزمة تشريح الجهاز الهيكلي (نظري 3)
💀💀💀💀💀💀💀💀💀💀
تتميز هذهِ الملزمة بعِدة مُميزات :
1- مُترجمة ترجمة تُناسب جميع المستويات
2- تحتوي على 78 رسم توضيحي لكل كلمة موجودة بالملزمة (لكل كلمة !!!!)
#فهم_ماكو_درخ
3- دقة الكتابة والصور عالية جداً جداً جداً
4- هُنالك بعض المعلومات تم توضيحها بشكل تفصيلي جداً (تُعتبر لدى الطالب أو الطالبة بإنها معلومات مُبهمة ومع ذلك تم توضيح هذهِ المعلومات المُبهمة بشكل تفصيلي جداً
5- الملزمة تشرح نفسها ب نفسها بس تكلك تعال اقراني
6- تحتوي الملزمة في اول سلايد على خارطة تتضمن جميع تفرُعات معلومات الجهاز الهيكلي المذكورة في هذهِ الملزمة
واخيراً هذهِ الملزمة حلالٌ عليكم وإتمنى منكم إن تدعولي بالخير والصحة والعافية فقط
كل التوفيق زملائي وزميلاتي ، زميلكم محمد الذهبي 💊💊
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This presentation was provided by Racquel Jemison, Ph.D., Christina MacLaughlin, Ph.D., and Paulomi Majumder. Ph.D., all of the American Chemical Society, for the second session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session Two: 'Expanding Pathways to Publishing Careers,' was held June 13, 2024.
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
2. Topic Outline
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
I. Characteristics of Mathematical Language
II. Expression versus Sentences
III. Conventions in the Mathematical Language
IV. Four Basic Concepts
V. Elementary Logic
VI. Formality
3. Conventions in the Mathematical Language
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
Mathematics is a spoken and written natural languages for
expressing mathematical language.
Mathematical language is an efficient and powerful tool for
mathematical expression, exploration, reconstruction after
exploration, and communication.
It is precise and concise.
It is has a poor understanding of the language.
4. Conventions in the Mathematical Language
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
Mathematics languages:
Digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9;
Mathematical symbols
5. Mathematical Language
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
Mathematical language is the system used to communicate
mathematical ideas.
It consists of some natural language using technical terms
(mathematical terms) and grammatical conventions that are
uncommon to mathematical discourse, supplemented by a
highly specialized symbolic notation for mathematical
formulas.
Mathematical notation used for formulas has its own grammar
and shared by mathematicians anywhere in the globe.
Mathematical language is being precise, concise, and powerful.
6. Expression versus Sentences
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
An expression (or mathematical expression) is a finite
combination of symbols that is well-defined according to rules
that depend on the context.
Symbols can designate numbers, variables, operations,
functions, brackets, punctuations, and groupings to help
determine order of operations, and other aspects of
mathematical syntax.
Expression – correct arrangement of mathematical symbols to
represent the object of interest, does not contain a complete
thought, and cannot be determined if it is true or false.
Some types of expressions are numbers, sets, and functions.
7. Expression versus Sentences
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
Sentence (or mathematical sentence) – a statement about two
expressions, either using numbers, variables, or a combination
of both.
Uses symbols or words like equals, greater than, or less than.
It is a correct arrangement of mathematical symbols that states
a complete thought and can be determined whether it’s true,
false, sometimes true/sometimes false.
8. Conventions in the Mathematical Language
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
Mathematical Convention is a fact, name, notation, or usage
which is generally agreed upon by mathematicians.
PEMDAS (Parenthesis, Exponent, Multiplication, Division,
Addition and Subtraction.)
All mathematical names and symbols are conventional.
9. Conventions in the Mathematical Language
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
Different and specific meaning within mathematics—
group ring field term factor,
Special terms—
tensor fractal function
Mathematical Taxonomy —
Axiom conjecture theorems lemma corollaries
10. Conventions in the Mathematical Language
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
Formulas are written predominantly left to right, even when
the writing system of the substrate language is right-to-left.
Mathematical expressions
= (equal) < (less-than) > (greater-than)
+ (addition) – (subtraction) (multiplication)
(division) (element) (for all)
(there exists) (infinity) (implies)
(if and only if) (approximately) (therefore)
Latin alphabet is commonly used for simple variables and
parameters.
11. Four Basic Concepts
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
A. Language of Sets
B. Language of Functions
C. Language of Relations
D. Language of Binary Operations
12. Language of Sets
Set theory is the branch of mathematics that studies sets or the
mathematical science of the infinite.
George Cantor (1845-1918) is a German
Mathematician
He is considered as the founder of set
theory as a mathematical discipline.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
13. Sets and Elements
A set is a well-defined collection of objects.
The objects are called the elements or members of the set.
element of a set
not an element of a set.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
14. Some Examples of Sets
D = {xx is an integer, 1 x 8}
A = {xx is a positive integer less than 10}
B = {xx is a real number and x2 – 1 = 0}
C = {xxis a letter in the word dirt}
E = {xx is a set of vowel letters}
Set E equals the set of all x such that x is a set of vowel
letters” or E = {a, e, i, o, u}
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
15. Indicate whether the ff. defined a Set
a. The list of course offerings of Centro Escolar University.
b. The elected district councilors of Manila City.
c. The collection of intelligent monkeys in Manila Zoo.
Answer:
Answer:
Answer:
Set
Set
Not a set
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
16. List the Elements of the Sets
a. A = {xx is a letter in the word mathematics.}
b. B = {xx is a positive integer, 3 x 8.}
c. C = {xx = 2n + 3, n is a positive integer.}
Answer:
Answer:
Answer:
A = {m, a, t, h, e, i, c, s.}
B = {3, 4, 5, 6, 7, 8}
C = {5, 7, 9, 11, 13, …}
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
17. Methods of Writing Sets
Roster Method. The elements of the set are enumerated and
separated by a comma it is also called tabulation method.
Rule Method. A descriptive phrase is used to describe the
elements or members of the set it is also called set builder
notation, symbol it is written as {x P(x)}.
Example:
E = {a, e, i, o, u}
E = {xx is a collection of vowel letters}
Roster method
Rule method
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
18. Write the ff. Sets in Roster Form
a. A= {xx is the letter of the word discrete}
b. B = {x3 x 8, x Z}
c. C = {xx is the set of zodiac signs}
Answer: A = {d, i, s, c, r, e, t}
B = {4, 5, 6, 7}
C = {Aries, Cancer, Capricorn, Sagittarius, Libra,
Leo, …}
Answer:
Answer:
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
19. Write the ff. Sets using Rule Method
a. D = {Narra, Mohagany, Molave, …}
c. F = {Botany, Biology, Chemistry, Physics, …}
b. E = {DOJ, DOH, DOST, DSWD, DENR, CHED, DepEd,…}
Answer: D = {xx is the set of non-bearing trees.}
E = {xx is the set of government agencies.}
F = {xx is the set of science subjects.}
Answer:
Answer:
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
20. Some Terms on Sets
Finite and Infinite Sets.
Unit Set
Cardinality
Empty Set
Universal Set
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
21. Finite Set
Finite set is a set whose elements are limited or countable, and
the last element can be identified.
Example:
c. E = {a, e, i, o, u}
b. C = {d, i, r, t}
a. A = {xx is a positive integer less than 10}
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
22. Infinite Set
Infinite set is a set whose elements are unlimited or
uncountable, and the last element cannot be specified.
Example:
c. H = {xx is a set of molecules on earth}
b. G = {xx is a set of whole numbers}
a. F = {…, –2, –1, 0, 1, 2,…}
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
23. Unit Set
A unit set is a set with only one element it is also called
singleton.
Example:
c. K = {rat}
b. J = {w}
a. I = {xx is a whole number greater than 1 but less than 3}
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
24. Empty Set
An empty set is a unique set with no elements (or null set), it is
denoted by the symbol or { }.
Example:
c. N = {xx is the set of positive integers less than zero}
b. M = {xx is a number of panda bear in Manila Zoo}
a. L = {xx is an integer less than 2 but greater than 1}
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
25. Universal Set
Universal set is the all sets under investigation in any
application of set theory are assumed to be contained in some
large fixed set, denoted by the symbol U.
Example:
c. U = {xx is an animal in Manila Zoo}
b. U = {1, 2, 3,…,100}
a. U = {xx is a positive integer, x2 = 4}
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
26. Cardinality
The cardinal number of a set is the number of elements or
members in the set, the cardinality of set A is denoted by n(A)
Example: Determine its cardinality of the ff. sets
c. C = {d, i, r, t}
b. A = {xx is a positive integer less than 10}
a. E = {a, e, i, o, u}, n(E) = 5
n(A) = 9
n(C) = 4
Answer
A = {1, 2, 3, 4, 5, 6, 7, 8, 9}
Theorem 1.1: Uniqueness of the Empty Set: There is only one set
with no elements.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
27. Venn Diagram
Venn Diagram is a pictorial presentation of relation and operations
on set.
Also known set diagrams, it show all hypothetically possible logical
relations between finite collections of sets.
Introduced by John Venn in his paper "On the Diagrammatic and
Mechanical Representation of Propositions and Reasoning’s"
Constructed with a collection of simple closed
curves drawn in the plane or normally
comprise of overlapping circles.
The interior of the circle symbolically
represents the elements (or members) of the
set, while the exterior represents elements
which are not members of the set.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
28. Kinds of Sets
Subset
Proper Subset
Equal Set
Power Set
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
29. Subset
If A and B are sets, A is called subset of B, if and only if, every
element of A is also an element of B.
Symbolically: A B x, x A x B.
Example: Suppose
A = {c, d, e}
B = {a, b, c, d, e}
U = {a, b, c, d, e, f, g}
Then A B, since all elements of A is in B.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
30. Proper Subset
Let A and B be sets. A is a proper subset of B, if and only if,
every element of A is in B but there is at least one element of B
that is not in A.
Symbolically: A B x, x A x B.
Example: Suppose
A = {c, d, e}
B = {a, b, c, d, e}
U = {a, b, c, d, e, f, g}
Then A B, since all elements of A is in B.
C = {e, a, c, b, d}
The symbol denotes that it is not a proper subset.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
31. Equal Sets
Given set A and B, A equals B, written, if and only if, every
element of A is in B and every element of B is in A.
Symbolically: A = B A B B A.
Example:
Suppose A = {a, b, c, d, e},
B = {a, b, d, e, c}
U = {a, b, c, d, e, f, g}
Then then A B and B A, thus A = B.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
32. Power Set
Given a set S from universe U, the power set of S denoted by
(S), is the collection (or sets) of all subsets of S.
Example: Determine the power set of (a) A = {e, f},
(b) = B = {1, 2, 3}.
(a) A = {e, f} (A) = {{e}, {f}, {e, f}, }
(b) B = {1, 2, 3} (B) = {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3},
{1, 2, 3}, }.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
33. Theorem
Theorem 1.2: A Set with No Elements is a Subset of Every Set: If
is a set with no elements and A is any set, then
A.
Theorem 1.3: For all sets A and B, if A B then (A) (B).
Theorem 1.4: Power Sets: For all integers n, if a set S has n
elements then (S) has 2n elements.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
34. Operations on Sets
Union
Intersection
Complement
Difference
Symmetric Difference
Disjoint Sets
Ordered Pairs
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
35. Union
The union of A and B, denoted AB, is the set of all elements x
in U such that x is in A or x is in B.
Symbolically: AB = {xx A x B}.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
36. Intersection
The intersection of A and B, denoted AB, is the set of all
elements x in U such that x is in A and x is in B.
Symbolically: AB = {xx A x B}.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
37. Complement
The complement of A (or absolute complement of A), denoted
A’, is the set of all elements x in U such that x is not in A.
Symbolically: A’ = {x U x A}.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
38. Difference
The difference of A and B (or relative complement of B with
respect to A), denoted A B, is the set of all elements x in U
such that x is in A and x is not in B.
Symbolically: A B = {xx A x B} = AB’.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
39. Symmetric Difference
If set A and B are two sets, their symmetric difference as the set
consisting of all elements that belong to A or to B, but not to
both A and B.
Symbolically: A B = {xx (AB) x(AB)}
= (AB)(AB)’ or (AB) (AB).
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
40. Example
Suppose
B = {c, d, e} U = {a, b, c, d, e, f, g}
A = {a, b, c}
Find the following
a. AB
b. AB
c. A’
d. A B
e. A B
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
41. Solution
a. AB
b. AB
d. A B
e. A B
= {a, b, c, d, e}
= {c}
= {a, b}
= {a, b, d, e}
c. A’ = {d, e, f, g}
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
42. Disjoint Sets
Two set are called disjoint (or non-intersecting) if and only if,
they have no elements in common.
Symbolically: A and B are disjoint AB = .
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
43. Order Pairs
In the ordered pair (a, b), a is called the first component and b is
called the second component. In general, (a, b) (b, a).
Example: Determine whether each statement is true or false.
a. (2, 5) = (9 – 7, 2 + 3)
b. {2, 5} {5, 2}
c. (2, 5) (5, 2)
Since 2 = 9 – 7 and 2 + 3 = 5, the ordered pair
is equal. True
Since these are sets and not ordered pairs,
the order in which the elements are listed is
not important. False
These ordered pairs are not equal since they
do not satisfy the requirements for equality
of ordered pairs. True
44. Cartesian Product
The Cartesian product of sets A and B, written AxB, is
AxB = {(a, b) a A and b B}
Example:
a. AxB
Let A = {2, 3, 5} and B = {7, 8}. Find each set.
b. BxA
c. AxA
= {(2, 7), (2, 8), (3, 7), (3, 8), (5, 7), (5, 8)}
= {(7, 2), (7, 3), (7, 5), (8, 2), (8, 3), (8, 5)}
= {(2, 2), (2, 3), (2, 5), (3, 2), (3, 3), (3, 5), (5, 2), (5, 3),
(5, 5)}
45. Language of Functions and Relations
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
A relation is a set of ordered pairs.
If x and y are elements of these sets and if a relation exists
between x and y, then we say that x corresponds to y or that y
depends on x and is represented as the ordered pair of (x, y).
A relation from set A to set B is defined to be any subset of AB.
If R is a relation from A to B and (a, b) R, then we say that “a
is related to b” and it is denoted as a R b.
46. Language of Functions and Relations
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
Let A = {a, b, c, d} be the set of car brands, and
B = {s, t, u, v} be the set of countries of the car manufacturer.
Then AB gives all possible pairings of the elements of A and B,
let the relation R from A to B be given by
R = {(a, s), (a, t), (a, u), (a, v), (b, s), (b, t), (b, u), (b, v), (c, s),
(c, t), (c, u), (b, v), (d, s), (d, t), (d, u), (d, v)}.
47. Language of Functions and Relations
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
Let R be a relation from set A to the set B.
domain of R is the set dom R
dom R = {a A (a, b) R for some b B}.
image (or range) of R
im R = {b B (a, b) R for some a A}.
Example: A = {4, 7},
Then AA = {(4, 4), (4, 7), (7, 4),(7, 7)}.
Let on A be the description of x y x + y is even.
Then (4, 4) , and (7, 7) .
48. Language of Functions and Relations
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
Function is a special kind of relation helps visualize
relationships in terms of graphs and make it easier to interpret
different behavior of variables..
Applications of Functions:
financial applications economics medicine
Engineering sciences natural disasters
calculating pH levels measuring decibels
designing machineries
49. Language of Binary Operations
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
A function is a relation in which, for
each value of the first component of
the ordered pairs, there is exactly one
value of the second component.
The set X is called the domain of the
function.
For each element of x in X, the corresponding element y in Y is
called the value of the function at x, or the image of x.
Range – set of all images of the elements of the domain is
called the of the function. A function can map from one set to
another.
50. Language of Binary Operations
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
Determine whether each of the following relations is a function.
A = {(1, 3), (2, 4), (3, 5), (4, 6)}
B = {(–2, 7), (–1, 3), (0, 1), (1, 5), (2, 5)}
C = {(3, 0), (3, 2), (7, 4), (9, 1)}
51. Language of Binary Operations
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
Algebraic structures focuses on investigating sets associated by
single operations that satisfy certain reasonable axioms.
An operation on a set generalized structures as the integers
together with the single operation of addition, or invertible 22
matrices together with the single operation of matrix
multiplication.
The algebraic structures known as group.
52. Binary Operations
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
Let G be a set. A binary operation on G is a function that assigns
each ordered pair of element of G.
Symbolically, a b = G, for all a, b, c G.
53. Group
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
A group is a set of elements, with one operation, that satisfies
the following properties:
(i) the set is closed with respect to the operation,
(ii) the operation satisfies the associative property,
(iii)there is an identity element, and
(iv) each element has an inverse.
54. Group
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
A group is an ordered pair (G, ) where G is a set and is a
binary operation on G satisfying the four properties.
Closure property. If any two elements are combined using
the operation, the result must be an element of the set.
a b = c G, for all a, b, c G.
Associative property. (a b) c = a (b c), for all a, b,
c G.
Identity property. There exists an element e in G, such that
for all a G, a e = e a.
Inverse property. For each a G there is an element a–1 of
G, such that a a–1 = a–1 a = e.
55. Group
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
The set of group G contain all the elements including the
binary operation result and satisfying all the four properties
closure, associative, identity e, and inverse a–1.
56. Example
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
Determine whether the set of all non-negative integers under
addition is a group.
Solution:
Apply the four properties to test the set of all non-negative
integers under addition is a group.
Step 1: Closure property, choose any two positive integers,
8 + 4 = 12 and 5 + 10 = 15
The sum of two numbers of the set, the result is always a
number of the set.
Thus, it is closed.
57. Solution
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
Step 2: Associative property, choose three positive integers
3 + (2 + 4) = 3 + 6 = 9
(3 + 2) + 4 = 5 + 4 = 9
Thus, it also satisfies the associative property.
Step 3: Identity property, choose any positive integer
8 + 0 = 8; 9 + 0 = 9; 15 + 0 = 15
Thus, it also satisfies the identity property.
58. Solution
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
Step 4: Inverse property, choose any positive integer
4 + (–4) = 0;
10 + (–10) = 0;
23 + (–23) = 0
Note that a–1 = –a.
Thus, it also satisfies the inverse property.
Thus, the set of all non-negative integers under addition is a
group, since it satisfies the four properties.
59. Formal Logic
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
The science or study of how to evaluate arguments &
reasoning.
It differentiate correct reasoning from poor reasoning.
It is important in sense that it helps us to reason correctly.
The methods of reasoning.
60. Mathematical Logic
Mathematical logic (or symbolic logic) is a branch of
mathematics with close connections to computer science.
Four Divisions:
Model Theory
Set Theory Recursion Theory
Proof Theory
It also study the deductive formal proofs systems and
expressive formal systems.
Mathematical study of logic and the applications of formal
logic to other areas of mathematics.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
61. Aristotle (382-322 BC)
Aristotle is generally regarded as
the Father of Logic
The study started in the late 19th
century with the development of
axiomatic frameworks for analysis,
geometry and arithmetic.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
62. Statement
A statement (or proposition) is a declarative sentence which
is either true or false, but not both.
The truth value of the statements is the truth and falsity of
the statement.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
63. Example
Which of the following are statements?
1. Manila is the capital of the Philippines. Is true
A statement.
2. What day is it? It is a question
Not a statement.
3. Help me, please. It cannot be categorized as true or false.
Not a statement.
4. He is handsome. Is neither true nor false - “he” is not
specified.
Not a statement.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
64. Ambiguous Statements
2. Calculus is more interesting than Trigonometry.
3. It was hot in Manila.
4. Street vendors are poor.
1. Mathematics is fun.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
65. Propositional Variable
A variable which used to represent a statement.
A formal propositional written using propositional logic
notation, p, q, and r are used to represent statements.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
66. Logical Connectives
A compound statement is a statement composed of two or
more simple statements connected by logical connectives
Logical connectives are used to combine simple statements
which are referred as compound statements.
A statement which is not compound is said to be simple
(also called atomic).
“exclusive-or.”
“or” “if then”
“and” “if and only if”
“not”
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
67. Conjunction
The conjunction of the statement p and q is the compound
statement “p and q.”
Symbolically, p q, where is the symbol for “and.”
Property 1: If p is true and q is true, then p q is true;
otherwise p q is false. Meaning, the conjunction of
two statements is true only if each statement is true.
p q p q
T
T
F
F
T
F
T
F
T
F
F
F
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
68. Example
1. 2 + 6 = 9 and man is a mammal.
Determine the truth value of each of the following conjunction.
2. Manny Pacquiao is a boxing champion and Gloria
Macapagal Arroyo is the first female Philippine
President.
3. Ferdinand Marcos is the only three-term Philippine
President and Joseph Estrada is the only Philippine
President who resigns.
False True
False
False
True
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
69. Statement
The disjunction of the statement p, q is the compound
statement “p or q.”
Property 2: If p is true or q is true or if both p and q are true,
then p q is true; otherwise p q is false. Meaning,
the disjunction of two statements is false only if
each statement is false.
Symbolically, p q, where is the symbol for “or.”
p q p q
T
T
F
F
T
F
T
F
T
T
T
F
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
70. Example
Determine the truth value of each of the following disjunction.
1. 2 + 6 = 9 or Manny Pacquiao is a boxing champion.
2. Joseph Ejercito is the only Philippine President who
resigns or Gloria Macapagal Arroyo is the first female
Philippine President.
3. Ferdinand Marcos is the only three-term Philippine
President or man is a mammal.
False True
True
True
True
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
71. Negation
The negation of the statement p is denoted by p, where is
the symbol for “not.”
Property 3: If p is true, p is false. Meaning, the truth value of
the negation of a statement is always the reverse of
the truth value of the original statement.
p p
T
F
F
T
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
72. Example
The following are statements for p, find the corresponding p.
1. 3 + 5 = 8.
2. Sofia is a girl.
3. Achaiah is not here.
3 + 5 8.
Sofia is a boy.
Achaiah is here.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
73. Conditional
The conditional (or implication) of the statement p and q is the
compound statement “if p then q.”
Symbolically, p q, where is the symbol for “if then.” p is called
hypothesis (or antecedent or premise) and q is called conclusion (or
consequent or consequence).
Property 4: The conditional statement
p q is false only when p is true and q is
false; otherwise p q is true. Meaning
p q states that a true statement cannot
imply a false statement.
p q p q
T
T
F
F
T
F
T
F
T
F
T
T
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
74. Example
In the statement “If vinegar is sweet, then sugar is sour.”
The antecedent is “vinegar is sweet,” and
the consequent is “sugar is sour.”
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
75. Example
Obtain the truth value of each of the following conditional
statements.
1. If vinegar is sweet, then sugar is sour.
2. 2 + 5 = 7 is a sufficient condition for 5 + 6 = 1.
3. 14 – 8 = 4 is a necessary condition that 6 3 = 2.
True
False
True
False False
True False
False True
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
76. Biconditional
The biconditional of the statement p and q is the compound
statement “p if and only if q.”
Property 5: If p and q are true or both false, then p q is true;
if p and q have opposite truth values, then p q is
false.
Symbolically, p q, where is the symbol for “if and only if.”
p q p q
T
T
F
F
T
F
T
F
T
F
F
T
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
77. Example
Determine the truth values of each of the following
biconditional statements.
1. 2 + 8 = 10 if and only if 6 – 3 = 3.
2. Manila is the capital of the Philippines is equivalent
to fish live in moon.
3. 8 – 2 = 5 is a necessary and sufficient for 4 + 2 = 7.
True
False
True
True True
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
78. Exclusive-Or
The exclusive-or of the statement p and q is the compound
statement “p exclusive or q.”
Property 6: If p and q are true or both false, then p q is false;
if p and q have opposite truth values, then p q is
true.
Symbolically, p q, where is the symbol for “exclusive or.”
p q p q
T
T
F
F
T
F
T
F
F
T
T
F
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
79. Example
“Sofia will take her lunch in Batangas or she will have it in
Singapore.”
Case 1: Sofia cannot have her lunch in Batangas and at
the same time in do it in Singapore,”
Case 2: If Sofia will have her lunch in Batangas or in
Singapore, meaning she can only have it in one
location given a single schedule.
Case 3: If she ought to decide to have her lunch
elsewhere (neither in Batangas nor in Singapore).
False
True
False
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
80. Predicate
A predicate (or open statements) is a statement whose truth
depends on the value of one or more variables.
Predicates become propositions once every variable is bound
by assigning a universe of discourse.
Most of the propositions are define in terms of predicates
“x is an even number” is a predicate whose truth depends
on the value of x.
Example:
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
81. Predicate
A predicate can also be denoted by a function-like notation.
P(x) = “x is an even number.” Now P(2) is true, and P(3)
is false.
If P is a predicate, then P(x) is either true or false, depending
on the value of x.
Example:
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
82. Propositional Function
A propositional function is a sentence P(x); it becomes a
statement only when variable x is given particular value.
Propositional functions are denoted as P(x), Q(x),R(x), and so
on.
The independent variable of propositional function must have
a universe of discourse, which is a set from which the variable
can take values.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
83. Propositional Function
“If x is an odd number, then x is not a multiple of 2.”
The given sentence has the logical form P(x) Q(x) and its
truth value can be determine for a specific value of x.
There exists an x such that x is odd number and 2x is even number.
Example:
For all x, if x is a positive integer, then 2x + 1 is an odd
number.
Example: Existential Quantifiers
Universal Quantifiers
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
84. Universe of Discourse
The universe of discourse for the variable x is the set of
positive real numbers for the proposition
“There exists an x such that x is odd number and 2x is even
number.”
Binding variable is used on the variable x, we can say that the
occurrence of this variable is bound.
A variable is said to be free, if an occurrence of a variable is
not bound.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
85. Universe of Discourse
To convert a propositional function into a proposition, all
variables in a proposition must be bound or a particular value
must be designated to them.
The scope of a quantifier is the part of an assertion in which
variables are bound by the quantifier.
This is done by applying combination of quantifiers
(universal, existential) and value assignments.
A variable is free if it is outside the scope of all quantifiers.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
86. Existential Quantifiers
The statement “there exists an x such that P(x),” is
symbolized by x P(x).
The symbol is called the existential quantifier
The statement “x P(x)”is true if there is at least one value of x
for which P(x) is true.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
87. Universal Quantifiers
The statement “for all x, P(x),” is symbolized by x P(x).
The symbol is called the universal quantifier.
The statement “x P(x)”is true if only if P(x) is true for every
value of x.
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
88. Topic Outline
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
Quantifier Symbol Translation
Existential There exists
There is some
For some
For which
For at least one
Such that
Satisfying
Universal For all
For each
For every
For any
Given any
89. Truth Values of Quantifiers
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
If the universe of discourse for P is P{p1, p2, …, pn}, then
x P(x) P(p1) P(p2) … P(pn) and
x P(x) P(p1) P(p2) … P(pn).
Statement Is True when Is False when
x P(x) P(x) is true for
every x.
There is at least
one x for which
P(x) is false.
x P(x) There is at least
one x for which
P(x) is true.
P(x) is false for
every x.
90. Quantified Statements and their Negation
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
Statement Negation
All A are B. Some A are not B.
No A are B. Some A are B.
Some A are not B. All A are B.
Some A are B. No A are B.