This slide help in the study of those students who are enrolled in BSCS BSC computer MSCS. In this slide introduction about discrete structure are explained. As soon as I upload my next lecture on proposition logic.
The document discusses common logarithms and natural logarithms. It defines common logarithms as logarithms with base 10, which are often used in real-world problems. It shows how to use properties of logarithms to solve exponential equations and by graphing. The document then defines natural logarithms as logarithms with base e, which are the inverse of exponential functions with base e. It demonstrates using natural logarithms and exponential base e to solve equations and expressions.
This document discusses sets, functions, and relations. It begins by defining key terms related to sets such as elements, subsets, operations on sets using union, intersection, difference and complement. It then discusses relations and functions, defining a relation as a set of ordered pairs where the elements are related based on a given condition. Functions are introduced as a special type of relation where each element of the domain has a single element in the range. The document aims to help students understand and use the properties and notations of sets, functions, and relations, and appreciate their importance in real-life situations.
The document defines basic concepts about sets including:
- A set is a collection of distinct objects called elements. Sets can be represented using curly brackets or the set builder method.
- Common set symbols are defined such as belongs to (∈), is a subset of (⊆), and is not a subset of (⊄).
- Types of sets like empty sets, singleton sets, finite sets, and infinite sets are described.
- Operations between sets such as union, intersection, difference, and complement are explained using Venn diagrams.
- Laws for sets like commutative, associative, distributive, double complement, and De Morgan's laws are listed.
- An example problem calculates
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.
The document discusses symmetric groups and permutations. Some key points:
- A symmetric group SX is the group of all permutations of a set X under function composition.
- Sn, the symmetric group of degree n, represents permutations of the set {1,2,...,n}.
- Permutations can be written using cycle notation, such as (1 2 3) or (1 4).
- Disjoint cycles commute; the product of two permutations is their composition as functions.
- Any permutation can be written as a product of disjoint cycles of length ≥2. Cycles of even length decompose into an odd number of transpositions, and vice versa for odd cycles.
This document discusses sets and operations on sets. Some key points covered include:
- Sets are collections of distinct elements that can be defined either by listing elements (roster method) or describing characteristics of elements (rule method).
- The cardinality of a set refers to the number of elements in the set. Two sets are equivalent if there is a one-to-one correspondence between their elements.
- Sets can be finite, containing a fixed number of elements, or infinite. Sets are equal if they contain identical elements and disjoint if they have no elements in common.
- Common set operations include intersection, union, difference, and complement. Intersection identifies elements shared by two sets while union combines all
The document discusses lines and planes in mathematics. It provides multiple ways to specify a line, including using two points, a point and slope, or a slope and y-intercept. Lines can also be described using vectors, with a line being the set of points a + tv, where a is a point on the line, v is a direction vector, and t is a real number. Planes are similarly defined as the set of points where the dot product of a normal vector p and the offset (x - a) is 0, where a is a point on the plane. An example shows how to check if three points lie on the same line by finding the line equation and checking if a third point satisfies it.
The document discusses common logarithms and natural logarithms. It defines common logarithms as logarithms with base 10, which are often used in real-world problems. It shows how to use properties of logarithms to solve exponential equations and by graphing. The document then defines natural logarithms as logarithms with base e, which are the inverse of exponential functions with base e. It demonstrates using natural logarithms and exponential base e to solve equations and expressions.
This document discusses sets, functions, and relations. It begins by defining key terms related to sets such as elements, subsets, operations on sets using union, intersection, difference and complement. It then discusses relations and functions, defining a relation as a set of ordered pairs where the elements are related based on a given condition. Functions are introduced as a special type of relation where each element of the domain has a single element in the range. The document aims to help students understand and use the properties and notations of sets, functions, and relations, and appreciate their importance in real-life situations.
The document defines basic concepts about sets including:
- A set is a collection of distinct objects called elements. Sets can be represented using curly brackets or the set builder method.
- Common set symbols are defined such as belongs to (∈), is a subset of (⊆), and is not a subset of (⊄).
- Types of sets like empty sets, singleton sets, finite sets, and infinite sets are described.
- Operations between sets such as union, intersection, difference, and complement are explained using Venn diagrams.
- Laws for sets like commutative, associative, distributive, double complement, and De Morgan's laws are listed.
- An example problem calculates
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.
The document discusses symmetric groups and permutations. Some key points:
- A symmetric group SX is the group of all permutations of a set X under function composition.
- Sn, the symmetric group of degree n, represents permutations of the set {1,2,...,n}.
- Permutations can be written using cycle notation, such as (1 2 3) or (1 4).
- Disjoint cycles commute; the product of two permutations is their composition as functions.
- Any permutation can be written as a product of disjoint cycles of length ≥2. Cycles of even length decompose into an odd number of transpositions, and vice versa for odd cycles.
This document discusses sets and operations on sets. Some key points covered include:
- Sets are collections of distinct elements that can be defined either by listing elements (roster method) or describing characteristics of elements (rule method).
- The cardinality of a set refers to the number of elements in the set. Two sets are equivalent if there is a one-to-one correspondence between their elements.
- Sets can be finite, containing a fixed number of elements, or infinite. Sets are equal if they contain identical elements and disjoint if they have no elements in common.
- Common set operations include intersection, union, difference, and complement. Intersection identifies elements shared by two sets while union combines all
The document discusses lines and planes in mathematics. It provides multiple ways to specify a line, including using two points, a point and slope, or a slope and y-intercept. Lines can also be described using vectors, with a line being the set of points a + tv, where a is a point on the line, v is a direction vector, and t is a real number. Planes are similarly defined as the set of points where the dot product of a normal vector p and the offset (x - a) is 0, where a is a point on the plane. An example shows how to check if three points lie on the same line by finding the line equation and checking if a third point satisfies it.
This document defines and provides examples of different types of sets: empty sets, singleton sets, finite and infinite sets, union of sets, intersection of sets, difference of sets, subset of a set, disjoint sets, and equality of two sets. Empty sets have no elements. Singleton sets contain one element. Finite sets have a predetermined number of elements while infinite sets may be countable or uncountable. The union of sets contains all elements that are in either set. The intersection contains elements common to both sets. The difference contains elements in the first set that are not in the second. A set is a subset if all its elements are also in another set. Sets are disjoint if their intersection is empty. Two sets are equal
This document introduces the concept of sets in discrete mathematics. It defines what a set is and discusses how sets are represented and notated. It also covers basic set operations like union and intersection. Some key points include:
- A set is a collection of distinct objects, called elements.
- Sets can be represented either by listing elements within curly braces or using set-builder notation describing a common property.
- Basic set operations include union, intersection, subset, power set, and the empty set. Union combines elements in sets while intersection finds elements common to sets.
A set is a collection of distinct objects called elements or members. A set can be defined using a roster method that lists the elements within curly brackets, or a rule method that describes a characteristic property that determines the elements. The cardinality of a set refers to the number of elements it contains, and can be finite or infinite.
This document defines and explains various concepts related to sets and relations. It discusses the four main set operations of union, intersection, complement, and difference. It then explains eight types of relations: empty, universal, identity, inverse, reflexive, symmetric, transitive, and equivalence relations. Finally, it defines partial ordering as a relation that is reflexive, antisymmetric, and transitive.
Introduction to Sets and Set Operations. The presentation include contents of a KWLH Chart, Essential Questions, and Self-Assessment Questions. With exploration and formative assessments.
This document provides an overview of functions and their graphs. It defines what constitutes a function, discusses domain and range, and how to identify functions using the vertical line test. Key points covered include:
- A function is a relation where each input has a single, unique output
- The domain is the set of inputs and the range is the set of outputs
- Functions can be represented by ordered pairs, graphs, or equations
- The vertical line test identifies functions as those where a vertical line intersects the graph at most once
- Intercepts occur where the graph crosses the x or y-axis
This document defines the angle of depression as the angle below the horizontal that an observer looks to see an object lower than them, and defines the angle of elevation as the angle above the horizontal an observer looks to see a higher object. It provides two examples using trigonometry: one calculating the height of a tree using a 51 degree angle of elevation, and another calculating the distance to a boat using a 10 degree angle of depression.
This document provides a summary and review of key concepts from Abstract Algebra Part 1, including:
- Groups and their properties of closure, associativity, identity, and inverses
- Examples of finite and infinite, abelian and non-abelian groups
- Subgroups, including cyclic subgroups, tests for subgroups, and examples
- Additional concepts like the order of an element, conjugation, and the center of a group
The document defines relations and functions. A relation is a set of ordered pairs where each element in the domain (set of x-values) is paired with an element in the range (set of y-values). A function is a special type of relation where each element of the domain is mapped to exactly one element in the range. The document provides examples of relations that are and are not functions based on this one-to-one mapping property. It also discusses using function notation and evaluating functions for different inputs. Finally, it explains how to determine the domain of a function by identifying values that would result in illegal operations like division by zero.
A set is an unordered collection of objects called elements. A set contains its elements. Sets can be finite or infinite. Common ways to describe a set include listing elements between curly braces or using a set builder notation. The cardinality of a set refers to the number of elements it contains. Power sets contain all possible subsets of a given set. Cartesian products combine elements of sets into ordered pairs. Common set operations include union, intersection, difference, and testing if sets are disjoint.
The document provides notes on group theory. It discusses the definition of groups and examples of groups such as (Z, +), (Q, ×), and Sn. Properties of groups like Lagrange's theorem and criteria for subgroups are also covered. The notes then discuss symmetry groups, defining isometries of R2 and showing that the set of isometries forms a group. Symmetry groups G(Π) of objects Π in R2 are introduced and shown to be subgroups. Specific examples of symmetry groups like those of triangles, squares, regular n-gons, and infinite strips are analyzed. Finally, the concept of group isomorphism is defined and examples are given to illustrate isomorphic groups.
Topology is the branch of mathematics concerned with properties that remain unchanged by deformations such as stretching or shrinking. It studies concepts like open sets, closed sets, limits, and neighborhoods. The product topology on X × Y has as its basis all sets of the form U × V, where U is open in X and V is open in Y. Projections map elements of a product space X × Y onto the first or second factor. The subspace topology on a subset Y of a space X contains all intersections of Y with open sets of X. The interior of a set A is the largest open set contained in A, while the closure of A is the smallest closed set containing A.
This document defines and provides examples of relations and functions. It explains that a relation connects elements between two or more sets, and provides examples of universal, identity, symmetric, inverse, reflexive, transitive, and equivalence relations. It then defines a function as a binary relation that associates every element in the first set to exactly one element in the second set. The document outlines the properties of one-to-one (injective), onto (surjective), and bijective (one-to-one and onto) functions, providing examples of each.
The document contains announcements and information about an exam for a class. It includes the following key points:
- Students should bring any grade-related questions about Exam 1 without delay. The homework for Exam 2 has been uploaded.
- The professor is planning to cover chapters 3, 5, and 6 for Exam 2.
- The last day for students to drop the class with a grade of "W" is February 4th.
Functions Representations
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 13, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
1) Set theory helps organize things into groups and understand logic. Key contributors include Georg Cantor, John Venn, George Boole, and Augustus DeMorgan.
2) A set is a collection of elements. A subset contains only elements that are also in another set. The cardinality of a set refers to the number of elements it contains.
3) Venn diagrams show relationships between sets using overlapping circles to represent their common elements.
The document discusses limits and continuity, explaining what limits are, how to evaluate different types of limits using techniques like direct substitution, dividing out, and rationalizing, and how limits relate to concepts like derivatives, continuity, discontinuities, and the intermediate value theorem. Special trig limits, properties of limits, and how limits can be used to find derivatives are also covered.
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.
This document provides notes on discrete mathematics. It begins by defining the two main types of mathematics: continuous mathematics, which involves real numbers and smooth curves, and discrete mathematics, which involves distinct, countable values between points.
The document then lists some common topics in discrete mathematics, such as sets, logic, graphs, and trees. It provides an overview of sets and set theory, including defining sets, representing sets, membership, important sets like natural and real numbers, and describing properties like cardinality, finite vs infinite sets, subsets, and empty/singleton/equal/equivalent sets. It also introduces basic set operations like union, intersection, difference, and Cartesian product.
This document provides an overview of basic discrete mathematical structures including sets, functions, sequences, sums, and matrices. It begins by defining a set as an unordered collection of elements and describes various ways to represent sets, such as listing elements or using set-builder notation. It then discusses operations on sets like unions, intersections, complements, and Cartesian products. Finally, it introduces functions as assignments of elements from one set to another. The document serves as an introduction to fundamental discrete structures used throughout mathematics.
This document defines and provides examples of different types of sets: empty sets, singleton sets, finite and infinite sets, union of sets, intersection of sets, difference of sets, subset of a set, disjoint sets, and equality of two sets. Empty sets have no elements. Singleton sets contain one element. Finite sets have a predetermined number of elements while infinite sets may be countable or uncountable. The union of sets contains all elements that are in either set. The intersection contains elements common to both sets. The difference contains elements in the first set that are not in the second. A set is a subset if all its elements are also in another set. Sets are disjoint if their intersection is empty. Two sets are equal
This document introduces the concept of sets in discrete mathematics. It defines what a set is and discusses how sets are represented and notated. It also covers basic set operations like union and intersection. Some key points include:
- A set is a collection of distinct objects, called elements.
- Sets can be represented either by listing elements within curly braces or using set-builder notation describing a common property.
- Basic set operations include union, intersection, subset, power set, and the empty set. Union combines elements in sets while intersection finds elements common to sets.
A set is a collection of distinct objects called elements or members. A set can be defined using a roster method that lists the elements within curly brackets, or a rule method that describes a characteristic property that determines the elements. The cardinality of a set refers to the number of elements it contains, and can be finite or infinite.
This document defines and explains various concepts related to sets and relations. It discusses the four main set operations of union, intersection, complement, and difference. It then explains eight types of relations: empty, universal, identity, inverse, reflexive, symmetric, transitive, and equivalence relations. Finally, it defines partial ordering as a relation that is reflexive, antisymmetric, and transitive.
Introduction to Sets and Set Operations. The presentation include contents of a KWLH Chart, Essential Questions, and Self-Assessment Questions. With exploration and formative assessments.
This document provides an overview of functions and their graphs. It defines what constitutes a function, discusses domain and range, and how to identify functions using the vertical line test. Key points covered include:
- A function is a relation where each input has a single, unique output
- The domain is the set of inputs and the range is the set of outputs
- Functions can be represented by ordered pairs, graphs, or equations
- The vertical line test identifies functions as those where a vertical line intersects the graph at most once
- Intercepts occur where the graph crosses the x or y-axis
This document defines the angle of depression as the angle below the horizontal that an observer looks to see an object lower than them, and defines the angle of elevation as the angle above the horizontal an observer looks to see a higher object. It provides two examples using trigonometry: one calculating the height of a tree using a 51 degree angle of elevation, and another calculating the distance to a boat using a 10 degree angle of depression.
This document provides a summary and review of key concepts from Abstract Algebra Part 1, including:
- Groups and their properties of closure, associativity, identity, and inverses
- Examples of finite and infinite, abelian and non-abelian groups
- Subgroups, including cyclic subgroups, tests for subgroups, and examples
- Additional concepts like the order of an element, conjugation, and the center of a group
The document defines relations and functions. A relation is a set of ordered pairs where each element in the domain (set of x-values) is paired with an element in the range (set of y-values). A function is a special type of relation where each element of the domain is mapped to exactly one element in the range. The document provides examples of relations that are and are not functions based on this one-to-one mapping property. It also discusses using function notation and evaluating functions for different inputs. Finally, it explains how to determine the domain of a function by identifying values that would result in illegal operations like division by zero.
A set is an unordered collection of objects called elements. A set contains its elements. Sets can be finite or infinite. Common ways to describe a set include listing elements between curly braces or using a set builder notation. The cardinality of a set refers to the number of elements it contains. Power sets contain all possible subsets of a given set. Cartesian products combine elements of sets into ordered pairs. Common set operations include union, intersection, difference, and testing if sets are disjoint.
The document provides notes on group theory. It discusses the definition of groups and examples of groups such as (Z, +), (Q, ×), and Sn. Properties of groups like Lagrange's theorem and criteria for subgroups are also covered. The notes then discuss symmetry groups, defining isometries of R2 and showing that the set of isometries forms a group. Symmetry groups G(Π) of objects Π in R2 are introduced and shown to be subgroups. Specific examples of symmetry groups like those of triangles, squares, regular n-gons, and infinite strips are analyzed. Finally, the concept of group isomorphism is defined and examples are given to illustrate isomorphic groups.
Topology is the branch of mathematics concerned with properties that remain unchanged by deformations such as stretching or shrinking. It studies concepts like open sets, closed sets, limits, and neighborhoods. The product topology on X × Y has as its basis all sets of the form U × V, where U is open in X and V is open in Y. Projections map elements of a product space X × Y onto the first or second factor. The subspace topology on a subset Y of a space X contains all intersections of Y with open sets of X. The interior of a set A is the largest open set contained in A, while the closure of A is the smallest closed set containing A.
This document defines and provides examples of relations and functions. It explains that a relation connects elements between two or more sets, and provides examples of universal, identity, symmetric, inverse, reflexive, transitive, and equivalence relations. It then defines a function as a binary relation that associates every element in the first set to exactly one element in the second set. The document outlines the properties of one-to-one (injective), onto (surjective), and bijective (one-to-one and onto) functions, providing examples of each.
The document contains announcements and information about an exam for a class. It includes the following key points:
- Students should bring any grade-related questions about Exam 1 without delay. The homework for Exam 2 has been uploaded.
- The professor is planning to cover chapters 3, 5, and 6 for Exam 2.
- The last day for students to drop the class with a grade of "W" is February 4th.
Functions Representations
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 13, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
1) Set theory helps organize things into groups and understand logic. Key contributors include Georg Cantor, John Venn, George Boole, and Augustus DeMorgan.
2) A set is a collection of elements. A subset contains only elements that are also in another set. The cardinality of a set refers to the number of elements it contains.
3) Venn diagrams show relationships between sets using overlapping circles to represent their common elements.
The document discusses limits and continuity, explaining what limits are, how to evaluate different types of limits using techniques like direct substitution, dividing out, and rationalizing, and how limits relate to concepts like derivatives, continuity, discontinuities, and the intermediate value theorem. Special trig limits, properties of limits, and how limits can be used to find derivatives are also covered.
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.
This document provides notes on discrete mathematics. It begins by defining the two main types of mathematics: continuous mathematics, which involves real numbers and smooth curves, and discrete mathematics, which involves distinct, countable values between points.
The document then lists some common topics in discrete mathematics, such as sets, logic, graphs, and trees. It provides an overview of sets and set theory, including defining sets, representing sets, membership, important sets like natural and real numbers, and describing properties like cardinality, finite vs infinite sets, subsets, and empty/singleton/equal/equivalent sets. It also introduces basic set operations like union, intersection, difference, and Cartesian product.
This document provides an overview of basic discrete mathematical structures including sets, functions, sequences, sums, and matrices. It begins by defining a set as an unordered collection of elements and describes various ways to represent sets, such as listing elements or using set-builder notation. It then discusses operations on sets like unions, intersections, complements, and Cartesian products. Finally, it introduces functions as assignments of elements from one set to another. The document serves as an introduction to fundamental discrete structures used throughout mathematics.
Set Theory - Unit -II (Mathematical Foundation Of Computer Science).pptxKalirajMariappan
The document provides an overview of basic concepts of sets and mathematical foundations of computer science. It defines what a set is, different ways to represent sets, important types of sets like finite, infinite, empty sets. It also discusses operations on sets like union, intersection, difference, complement and Cartesian product. Finally, it covers concepts like cardinality, countable and uncountable sets, relations and different types of relations.
A power point presentation on the topic SETS of class XI Mathematics. it includes all the brief knowledge on sets like their intoduction, defination, types of sets with very intersting graphics n presentation.
After going through this module, you are expected to:
• define well-defined sets and other terms associated to sets
• write a set in two different forms;
• determine the cardinality of a set;
• enumerate the different subsets of a set;
• distinguish finite from infinite sets; equal sets from equivalent sets
• determine the union, intersection of sets and the difference of two sets
This document provides an overview of set theory, including definitions and concepts. It begins by defining a set as a collection of distinct objects, called elements or members. It describes how sets are denoted and provides examples. Key concepts covered include subsets, the empty set, set operations like union and intersection, and properties of sets. The document also discusses topics like the power set, Cartesian products, partitions, and the universal set. Overall, it serves as a comprehensive introduction to the basic ideas and terminology of set theory.
1. A set is a collection of distinct objects or elements that have some common property. Sets are represented using curly braces and capital letters. Elements within a set should not be repeated.
2. There are two main methods to define a set - the roster method which lists all elements, and the set builder method which describes the property defining membership.
3. There are several types of sets including finite sets with a set number of elements, infinite sets, singleton sets with one element, empty sets with no elements, equal sets with the same elements, and subsets which are sets within other sets.
Set and Set operations, UITM KPPIM DUNGUNbaberexha
This document defines sets and common set operations such as union, intersection, difference, complement, Cartesian product, and cardinality. It begins by defining a set as a collection of distinct objects and provides examples of sets. It then discusses ways to represent and visualize sets using listings, set-builder notation, Venn diagrams, and properties of subsets, supersets, equal sets, disjoint sets, and infinite sets. The document concludes by defining common set operations and identities using membership tables and examples.
This document provides an introduction to set theory. It defines what a set is and provides examples of common sets. A set can be represented in roster form by listing its elements within curly brackets or in set builder form using a characteristic property. There are different types of sets such as finite sets, infinite sets, empty sets, singleton sets, and power sets which contain all subsets. Set operations like union, intersection, difference and symmetric difference are introduced. Important concepts like subsets, equivalent sets, disjoint sets and complements are also covered.
This document defines and explains key concepts in sets and set theory, including:
- Defining sets and listing elements using roster and set-builder notation
- Important mathematical sets like natural numbers, integers, and real numbers
- Describing relationships between sets such as subsets, equality, and Venn diagrams
- Calculating cardinality to determine the number of elements in a set
- Forming Cartesian products to combine elements from multiple sets into ordered pairs
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 defines and describes several key concepts in set theory:
1. A set is a collection of well-defined objects that can be clearly distinguished from one another. Examples of sets include the set of natural numbers from 1 to 50.
2. Sets can be described using either the roster method, which lists the elements within curly brackets, or the set-builder method, which defines a property for the elements.
3. Types of sets include the empty set, singleton sets containing one element, finite sets that can be counted, infinite sets that cannot be counted, equivalent sets with the same number of elements, and subsets where all elements of one set are contained within another set.
The document introduces basic concepts of set theory, including:
- A set is a collection of distinct objects called elements or members.
- Special sets include the natural numbers, integers, rational numbers, and real numbers.
- Types of sets include subsets, equal sets, empty sets, singleton sets, finite sets, infinite sets, disjoint sets, power sets, and universal sets.
- Cardinal numbers represent the number of elements in a set.
A set is an unordered collection of unique elements. A set can be written explicitly using set brackets. The order of elements in a set does not matter. Some key concepts about sets discussed in the document include: types of sets like finite, empty, singleton, infinite sets; cardinality which is the number of elements in a set; set operations like intersection, union, complement; and set relations which describe connections between elements of different sets using ordered pairs. Common relations include empty, full, identity, inverse, symmetric, transitive, and equivalence relations.
Discrete mathematics for diploma studentsZubair Khan
This document discusses sets and set operations. It defines what a set is and how elements of sets are denoted. It describes ways to represent sets, such as listing elements or using set-builder notation. It discusses operations on sets like union, intersection, difference, and complement. It provides examples of applying these set concepts and operations. Exercises are included for the reader to practice working with sets.
This document provides information about sets and operations on sets. It begins by introducing sets and their representation using roster or tabular form and set builder form. It then defines different types of sets such as empty, singleton, finite, and infinite sets. It also discusses subsets, intervals as subsets of real numbers, the universal set, and the power set. Finally, it describes set operations like union and intersection and their properties.
The document defines sets and provides examples of different types of sets. It discusses how sets can be defined using roster notation by listing elements within braces or using set-builder notation stating properties elements must have. Some key types of sets mentioned include the empty set, natural numbers, integers, rational numbers, and real numbers. Operations on sets like union, intersection, and difference are introduced along with examples. Subsets, Venn diagrams, and the power set are also covered.
The document provides information about sets, relations, and functions in mathematics:
- A set is a collection of distinct objects, called elements or members. Sets are represented using curly brackets and elements are separated by commas. There are finite and infinite sets. Operations on sets include union, intersection, complement, difference, and power set.
- A relation from a set A to a set B is a subset of the Cartesian product A × B. The domain is the set of first elements in the relation and the range is the set of second elements.
- A function from a set A to a set B is a special type of relation where each element of A is mapped to exactly one element of B.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
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Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
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Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
4. Discrete Mathematics
Discrete is not the name of branch of mathematics like number theory, algebra ,calculus Rather it
is description of a set branches of math that all have the common feature that they are “discrete”
rather than “continues”
5. Discrete Mathematics
Mathematics can be broadly classified into two categories:
Continuous Mathematics ─ It is based upon continuous number line or the real numbers. It is
characterized by the fact that between any two numbers, there are almost always an infinite set of
numbers. For example, a function in continuous mathematics can be plotted in a smooth curve
without breaks.
Discrete Mathematics ─ It involves distinct values; i.e. between any two points, there are a
countable number of points. For example, if we have a finite set of objects, the function can be
defined as a list of ordered pairs having these objects, and can be presented as a complete list of
those pairs.
6. Discrete Mathematics
Topics in Discrete Mathematics
Though there cannot be a definite number of branches of Discrete Mathematics, the following topics
are almost always covered in any study regarding this matter:
Sets, Relations and Functions
Mathematical Logic
Group theory
Counting Theory
Probability
Mathematical Induction and Recurrence Relations
Graph Theory
Trees
Boolean Algebra
8. SET
A set is an unordered collection of different elements. A set can be written
explicitly by listing its elements using set bracket. If the order of the elements is
changed or any element of a set is repeated, it does not make any changes in the
set.
Or
Set can be defined as collection of distinct objects of any sort
9. SET
Example :
The sets of poets in Pakistan
The set of all ideas contained in this book
A collection of rocks
A bouquet of flower
Sets of Professors in Northern University
Generally speaking we think that set is collection of objects which share some
common properties.
10. SET
Some special sets
Set of natural numbers:
Letter ‘N’ is used to show the set of Natural Number
N = {0,1,2,3,4,……..}
Set of integer:
Letter ‘Z’ is used to indicate the set of integer
Z = {……….,-3,-2,-1,0,1,2,3,…………}
Set of Positive Number:
Letter ‘Z’ is used to show the set of Positive numbers
Z = {1,2,3,4,……}
11. SET
Empty Set
A set without any member is called empty set
It is denoted by {} or Ø
12. SET
Countable and uncountable set
In a finite set A we can always designate 1 element as a1 and 2 element as a2 and
so forth and if there are k elements in the set then these elements can be listed in
the following order is called countable set
Example :
a1,a2,a3,a4,………..,ak
If the set is infinite we may be able to select the 1 element as a1 and 2 element as
a2 and so on. So that the list
a1,a2,a3,a4,………..
13. SET
Representation of a Set Sets can be represented in two ways:
Tabular Form
Set Builder Notation
14. SET
Set builder form
Roster or Tabular Form: The set is represented by listing all the elements comprising it.
The elements are enclosed within braces and separated by commas.
Example 1: Set of vowels in English alphabet, A = {a,e,i,o,u}
Example 2: Set of odd numbers less than 10, B = {1,3,5,7,9}
15. SET
Set Builder Notation:
The set is defined by specifying a property that elements of the set have in
common.
The set is described as A = { x : p(x)}
A = { x : 1 <= x <= 4 } or A = {x | x is integer, x<8}
Example :
A = {2,3,4,5,6,……………,99}
A = { x:2 <= x<=99}
16. SET
Cardinality of a set
The number of elements in a set is called its cardinality. This is done by placing the vertical bars around
the set.
Or
No of distinct elements in set
Example:
A= {2,4,6,8}
|A| = |{2,4,6,8}| = 4
|B| = |{1, 2, 3,4,5,…}| = ∞
|C| = |{1,2,2,2,2}| = 2
17. SET
Types of Sets
Sets can be classified into many types. Some of which are finite, infinite, subset,
universal, proper, singleton set, etc.
Finite Set A set which contains a definite number of elements is called a finite
set.
Example: S = {x | x ∈ N and 70 > x > 50}
Infinite Set A set which contains infinite number of elements is called an
infinite set.
Example: S = {x | x ∈ N and x > 10}
18. SET
Some important definitions
Subset:
A set X is a subset of set Y (Written as X ⊆ Y) if every element of X is an
element of set Y.
Example 1: Let, X = { 1, 2, 3, 4, 5, 6 } and Y = { 1, 2 }. Here set Y is a subset of
set X as all the elements of set Y is in set X. Hence, we can write Y ⊆ X.
Example 2: Let, X = {1, 2, 3} and Y = {1, 2, 3}. Here set Y is a subset (Not a
proper subset) of set X as all the elements of set Y is in set X. Hence, we can
write Y ⊆ X.
19. SET
Proper Subset
The term “proper subset” can be defined as “subset of but not equal to”. A Set X
is a proper subset of set Y (Written as X ⊂ Y) if every element of X is an element
of set Y and
| X| < | Y |.
Example: Let, X = {1, 2, 3, 4, 5, 6} and Y = {1, 2}. Here set Y ⊂ X since all elements
in Y are contained in X too and X has at least one element is more than set Y.
20. SET
Universal Set
It is a collection of all elements in a particular context or application. All the sets
in that context or application are essentially subsets of this universal set.
Universal sets are represented as U.
Example: We may define U as the set of all animals on earth. In this case, set of
all mammals is a subset of U, set of all fishes is a subset of U, set of all insects is
a subset of U, and so on.
21. SET
Singleton Set or Unit Set
Singleton set or unit set contains only one element. A singleton set is denoted by {s}.
Example: S = {x | x ∈ N, 7 < x < 9} = { 8 }
22. SET
Equal Set If two sets contain the same elements they are said to be equal.
Example: If A = {1, 2, 6} and B = {6, 1, 2}, they are equal as every element of
set A is an element of set B and every element of set B is an element of set A.
23. SET
Equivalent Set If the cardinalities of two sets are same, they are called
equivalent sets.
Example: If A = {1, 2, 6} and B = {16, 17,17, 22}, they are equivalent as
cardinality of A is equal to the cardinality of B. i.e. |A|=|B|=3
24. SET
Overlapping Set
Two sets that have at least one common element are called overlapping sets.
Example: Let, A = {1, 2, 6} and B = {6, 12, 42}. There is a common element ‘6’,
hence these sets are overlapping sets
25. SET
Disjoint Set
Two sets A and B are called disjoint sets if they do not have even one element in
common. Therefore, disjoint sets have the following properties:
Example: Let, A = {1, 2, 6} and B = {7, 9, 14}; there is not a single common
element, hence these sets are overlapping sets.
26. SET
Empty Set or Null Set
An empty set contains no elements. It is denoted by ∅. As the number of
elements in an empty set is finite, empty set is a finite set. The cardinality of
empty set or null set is zero.
Example: S = {x | x ∈ N and 7 < x < 8} = ∅
31. Venn Diagram
Set Union
The union of sets A and B (denoted by A ∪ B) is the set of elements which are in A, in B,
or in both A and B. Hence, A∪B = {x | x ∈A OR x ∈B}.
Example: If A = {10, 11, 12, 13} and B = {13, 14, 15}, then A ∪ B = {10, 11, 12, 13,
14, 15}. (The common element occurs only once)
32. Venn Diagram
Set Intersection
The intersection of sets A and B (denoted by A ∩ B) is the set of elements which are in
both A and B. Hence, A∩B = {x | x ∈AAND x ∈B}.
Example: If A = {11, 12, 13} and B = {13, 14, 15}, then A∩B = {13}.
33. Venn Diagram
Set Difference/ Relative Complement
The set difference of sets A and B (denoted by A–B) is the set of elements which are only
in A but not in B. Hence, A−B = {x | x ∈AAND x ∉B}.
Example: If A = {10, 11, 12, 13} and B = {13, 14, 15}, then (A−B) = {10, 11, 12} and
(B−A) = {14,15}. Here, we can see (A−B) ≠ (B−A)
34. Venn Diagram
Complement of a Set
The complement of a set A (denoted by A’) is the set of elements which are not in set A. Hence, A'
= {x | x ∉A}.
More specifically, A'= (U–A) where U is a universal set which contains all objects.
Example: If A ={x | x belongs to set of odd integers} then A' ={y | y does not belong to set of odd
integers}
35. Venn Diagram
Cartesian Product / Cross Product
The Cartesian product of n number of sets A1, A2.....An, denoted as A1 × A2 ×..... × An,
can be defined as all possible ordered pairs (x1,x2,....xn) where x1∈ A1 , x2∈ A2 , ...... xn ∈ An
Example: If we take two sets A= {a, b} and B= {1, 2},
The Cartesian product of A and B is written as: A×B= {(a, 1), (a, 2), (b, 1), (b, 2)}
The Cartesian product of B and A is written as: B×A= {(1, a), (1, b), (2, a), (2, b)}
36. Venn Diagram
Power Set
Power set of a set S is the set of all subsets of S including the empty set. The cardinality
of a power set of a set S of cardinality n is 2n. Power set is denoted as P(S).
37. Example:
For a set S = {a, b, c, d} let us calculate the subsets:
Subsets with 0 elements: {∅} (the empty set)
Subsets with 1 element: {a}, {b}, {c}, {d}
Subsets with 2 elements: {a,b}, {a,c}, {a,d}, {b,c}, {b,d},{c,d}
Subsets with 3 elements: {a,b,c},{a,b,d},{a,c,d},{b,c,d}
Subsets with 4 elements: {a,b,c,d}
Hence, P(S) =
{ {∅},{a}, {b}, {c}, {d},{a,b}, {a,c}, {a,d}, {b,c},
{b,d},{c,d},{a,b,c},{a,b,d},{a,c,d},{b,c,d},{a,b,c,d} }
| P(S) | = 16
Note: The power set of an empty set is also an empty set.
| P ({∅}) | = 1
38. Venn Diagram
Partitioning of a Set
Partitioning of a Set Partition of a set, say S, is a collection of n disjoint subsets, say P1, P2,...…
Pn, that satisfies the following three conditions:
Pi does not contain the empty set. [ Pi ≠ {∅} for all 0 < i ≤ n]
The union of the subsets must equal the entire original set. [P1 ∪ P2 ∪ .....∪ Pn = S]
The intersection of any two distinct sets is empty. [Pa ∩ Pb ={∅}, for a ≠ b where n ≥ a, b ≥ 0 ]
Example Let S = {a, b, c, d, e, f, g, h}
One probable partitioning is {a}, {b, c, d}, {e, f, g,h}
Another probable partitioning is {a,b}, { c, d}, {e, f, g,h}
39. Venn Diagram
Bell Numbers
Bell numbers give the count of the number of ways to partition a set. They are denoted by Bn
where n is the cardinality of the set.
Example:
Let S = { 1, 2, 3}, n = |S| = 3
The alternate partitions are: 1. ∅, {1, 2, 3} 2. {1}, {2, 3} 3. {1, 2}, {3} 4. {1, 3}, {2} 5. {1},
{2},{3}
Hence B3 = 5
42. Relations
Relations may exist between objects of the same set or between objects of two or more sets.
Definition and Properties: A binary relation R from set x to y (written as xRy or R(x,y)) is a
subset of the Cartesian product x × y. If the ordered pair of G is reversed, the relation also
changes.
Generally an n-ary relation R between sets A1, ... , and An is a subset of the n-ary product
A1×...×An. The minimum cardinality of a relation R is Zero and maximum is n2 in this case.
A binary relation R on a single set A is a subset of A × A.
For two distinct sets, A and B, having cardinalities m and n respectively, the maximum cardinality
of a relation R from A to B is mn.
43. Relations
Domain and Range
If there are two sets A and B, and relation R have order pair (x, y), then:
Examples
Let, A = {1,2,9} and B = {1,3,7}
Case 1:
If relation R is ‘equal to’ then R = {(1, 1), (3, 3)} Dom(R) = { 1, 3}, Ran(R) = { 1, 3}
Case 2:
If relation R is ‘less than’ then R = {(1, 3), (1, 7), (2, 3), (2, 7)} Dom(R) = { 1, 2}, Ran(R) = { 3, 7}
Case 3:
If relation R is ‘greater than’ then R = {(2, 1), (9, 1), (9, 3), (9, 7)} Dom(R) = { 2, 9}, Ran(R) = { 1, 3, 7}
44. Relations
Representation of Relations using Graph
A relation can be represented using a directed graph.
The number of vertices in the graph is equal to the number of elements in the set from which the
relation has been defined. For each ordered pair (x, y) in the relation R, there will be a directed
edge from the vertex ‘x’ to vertex ‘y’. If there is an ordered pair (x, x), there will be self- loop on
vertex ‘x’.
Suppose, there is a relation R = {(1, 1), (1,2), (3, 2)} on set S = {1,2,3}, it can be represented by
the following graph:
45. Relations
Types of Relations
1. The Empty Relation between sets X and Y, or on E, is the empty set ∅
2. The Full Relation between sets X and Y is the set X×Y
3. The Identity Relation on set X is the set {(x,x) | x ∈ X}
4. The Inverse Relation R' of a relation R is defined as: R’= {(b,a) | (a,b) ∈R}
Example: If R = {(1, 2), (2,3)} then R’ will be {(2,1), (3,2)}
5. A relation R on set A is called Reflexive if ∀a∈A is related to a (aRa holds).
Example: The relation R = {(a,a), (b,b)} on set X={a,b} is reflexive
46. Relations
6. A relation R on set A is called Irreflexive if no a∈A is related to a (aRa does not hold). Example:
The relation R = {(a,b), (b,a)} on set X={a,b} is irreflexive
7. A relation R on set A is called Symmetric if xRy implies yRx, ∀x∈A and ∀y∈A.
Example: The relation R = {(1, 2), (2, 1), (3, 2), (2, 3)} on set A={1, 2, 3} is symmetric.
8. A relation R on set A is called Anti-Symmetric if xRy and yRx implies x=y ∀x ∈ A and ∀y ∈ A.
Example: The relation R = { (x,y) ∈ N | x ≤ y } is anti-symmetric since x ≤ y and y ≤ x implies x = y.
9. A relation R on set A is called Transitive if xRy and yRz implies xRz, ∀x,y,z ∈ A.
Example: The relation R = {(1, 2), (2, 3), (1, 3)} on set A= {1, 2, 3} is transitive.
10. A relation is an Equivalence Relation if it is reflexive, symmetric, and transitive.
Example: The relation R = {(1, 1), (2, 2), (3, 3), (1, 2),(2,1), (2,3), (3,2), (1,3), (3,1)} on set A= {1, 2,
3} is an equivalence relation since it is reflexive, symmetric, and transitive.
49. Function
A function or mapping (Defined as f: X→Y) is a relationship from elements of one set X to
elements of another set Y (X and Y are non-empty sets). X is called Domain and Y is called
Codomain of function ‘f’.
Function ‘f’ is a relation on X and Y such that for each x ∈ X, there exists a unique y ∈ Y
such that (x,y) ∈ R. ‘x’ is called pre-image and ‘y’ is called image of function f.
A function can be one to one or many to one but not one to many.
50. Function
Injective / One-to-one function
A function f: A→B is injective or one-to-one function if for every b ∈ B, there exists at most
one a ∈ A such that f(s) = t.
This means a function f is injective if a1 ≠ a2 implies f(a1) ≠ f(a2).
Example:
4. The function f : [3] → {a, b, c, d} defined by f(1) = c, f(2) = b and f(3) = a, is one-one.
Verify that there are 24 one-one functions f : [3] → {a, b, c, d}.
6. There is no one-one function from the set [3] to its proper subset [2].
7. There are one-one functions f from the set N to its proper subset {2, 3, . . .}. One of them
is given by f(1) = 3, f(2) = 2 and f(n) = n + 1, for n ≥ 3.
51. Function
Surjective / Onto function
A function f: A →B is surjective (onto) if the image of f equals its range. Equivalently, for
every b ∈ B, there exists some a ∈ A such that f(a) = b. This means that for any y in B,
there exists some x in A such that y = f(x).
Example
1. f : N→N, f(x) = x + 2 is surjective.
2. f : R→R, f(x) = x2 is not surjective since we cannot find a real number whose square
is negative.
52. Function
Bijective / One-to-one Correspondent
A function f: A →B is bijective or one-to-one correspondent if and only if f is both injective
and surjective.
Problem:
Prove that a function f: R→R defined by f(x) = 2x – 3 is a bijective function.
Explanation: We have to prove this function is both injective and surjective.
If f(x1) = f(x2), then 2x1 – 3 = 2x2 – 3 and it implies that x1 = x2.
Hence, f is injective.
Here, 2x – 3= y
So, x = (y+5)/3 which belongs to R and f(x) = y.
Hence, f is surjective.
Since f is both surjective and injective, we can say f is bijective.
53. Function
Inverse of a Function
The inverse of a one-to-one corresponding function f : A -> B, is the function g : B -> A, holding
the following property:
f(x) = y g(y) = x
The function f is called invertible, if its inverse function g exists. Example:
A function f : Z Z, f(x) = x + 5, is invertible since it has the inverse function
g : Z -> Z, g(x) = x – 5
A function f : Z -> Z, f(x) = x2 is not invertible since this is not one-to-one as (-x)2 = x2.
54. Function
Composition of Functions
Two functions f: A→B and g: B→C can be composed to give a composition g o f. This is a
function from A to C defined by (gof)(x) = g(f(x))
Example:
Let f(x) = x + 2 and g(x) = 2x + 1, find ( f o g)(x) and ( g o f)(x)
Solution (f o g)(x) = f (g(x)) = f(2x + 1) = 2x + 1 + 2 = 2x + 3
(g o f)(x) = g (f(x)) = g(x + 2) = 2 (x+2) + 1 = 2x + 5
Hence, (f o g)(x) ≠ (g o f)(x)
55. Function
Some Facts about Composition
If f and g are one-to-one then the function (g o f) is also one-to-one.
If f and g are onto then the function (g o f) is also onto.
Composition always holds associative property but does not hold commutative property.