This document introduces sets and their representations. It discusses:
1) Georg Cantor developed the theory of sets in the late 19th century while working on trigonometric series. Sets are now fundamental in mathematics.
2) A set is a well-defined collection of objects where we can determine if an object belongs to the set or not. Sets are represented using roster form (listing elements between braces) or set-builder form (using properties of elements).
3) The empty set, denoted {}, is the set with no elements. It is different from non-existence of a set.
The document defines and provides examples of different types of numbers:
1) Natural numbers are positive integers like 1, 2, 3. Whole numbers include natural numbers and 0.
2) Integers include all whole numbers and their negatives. Rational numbers are numbers that can be expressed as fractions. Irrational numbers cannot be expressed as fractions.
3) Real numbers include all rational and irrational numbers and can be represented on a number line. Some key rational numbers are terminating and non-terminating decimals. The Pythagoreans discovered irrational numbers like √2.
The document discusses complex numbers. It explains that the equation x^2 = -1 has no real solutions, so an imaginary number i is defined where i^2 = -1. A complex number is defined as a + bi, where a is the real part and bi is the imaginary part. Complex numbers can be added or multiplied by treating i as a variable and using rules like i^2 = -1. Examples show how to solve equations and perform operations with complex numbers.
This document provides a review of mathematics concepts for grade six students. It covers the history of numbers and various numeration systems used throughout history such as tally marks, Egyptian hieroglyphics, Babylonian, Greek, Chinese, Roman, and Hindu-Arabic systems. It also reviews concepts of place value, operations on whole numbers and decimals, word problems, and the order of operations. The material aims to help students understand and apply mathematical concepts.
The document discusses evaluating the formula log[(2x+1)/(sin1/3(x)+1)] at x=0 and x=10 degrees using a scientific calculator. It explains that the answer is 0 at x=0 and approximately 1.13 at x=10 degrees. It then describes the keyboard of a typical scientific calculator, noting the number, operation, yx, sin, log and formula keys. The rest of the document provides examples and definitions of algebraic, trigonometric and exponential-log formulas.
The document provides examples and explanations of adding, subtracting, multiplying, and expanding polynomials. It demonstrates multiplying polynomials using the FOIL (First, Outer, Inner, Last) method and provides examples of sum and difference of squares, square of a binomial, cube of a binomial, and multiplying three binomials. Common patterns that arise when multiplying polynomials are identified.
This document provides a lesson on complex numbers and using imaginary numbers to solve quadratic equations with complex roots. It begins with examples of simplifying expressions involving square roots of negative numbers by expressing them in terms of the imaginary unit i. Then it shows how to solve quadratic equations with imaginary solutions by taking square roots and expressing the solutions in terms of i. Examples are also provided of finding the complex zeros of quadratic functions by setting the functions equal to 0 and factoring. The document emphasizes that complex solutions come in conjugate pairs, and gives examples of finding the complex conjugate of solutions.
The document describes the number line and how it assigns numbers to points on a line. It explains that 0 is assigned to the center or origin of the line. Positive numbers are assigned to points to the right of 0, and negative numbers to points on the left. The number line defines the relative size of numbers based on their position, with numbers to the right being greater than those to the left. Intervals on the number line, such as -1 < x < 3, represent all the numbers between and including the bounds.
The document defines and provides examples of different types of numbers:
1) Natural numbers are positive integers like 1, 2, 3. Whole numbers include natural numbers and 0.
2) Integers include all whole numbers and their negatives. Rational numbers are numbers that can be expressed as fractions. Irrational numbers cannot be expressed as fractions.
3) Real numbers include all rational and irrational numbers and can be represented on a number line. Some key rational numbers are terminating and non-terminating decimals. The Pythagoreans discovered irrational numbers like √2.
The document discusses complex numbers. It explains that the equation x^2 = -1 has no real solutions, so an imaginary number i is defined where i^2 = -1. A complex number is defined as a + bi, where a is the real part and bi is the imaginary part. Complex numbers can be added or multiplied by treating i as a variable and using rules like i^2 = -1. Examples show how to solve equations and perform operations with complex numbers.
This document provides a review of mathematics concepts for grade six students. It covers the history of numbers and various numeration systems used throughout history such as tally marks, Egyptian hieroglyphics, Babylonian, Greek, Chinese, Roman, and Hindu-Arabic systems. It also reviews concepts of place value, operations on whole numbers and decimals, word problems, and the order of operations. The material aims to help students understand and apply mathematical concepts.
The document discusses evaluating the formula log[(2x+1)/(sin1/3(x)+1)] at x=0 and x=10 degrees using a scientific calculator. It explains that the answer is 0 at x=0 and approximately 1.13 at x=10 degrees. It then describes the keyboard of a typical scientific calculator, noting the number, operation, yx, sin, log and formula keys. The rest of the document provides examples and definitions of algebraic, trigonometric and exponential-log formulas.
The document provides examples and explanations of adding, subtracting, multiplying, and expanding polynomials. It demonstrates multiplying polynomials using the FOIL (First, Outer, Inner, Last) method and provides examples of sum and difference of squares, square of a binomial, cube of a binomial, and multiplying three binomials. Common patterns that arise when multiplying polynomials are identified.
This document provides a lesson on complex numbers and using imaginary numbers to solve quadratic equations with complex roots. It begins with examples of simplifying expressions involving square roots of negative numbers by expressing them in terms of the imaginary unit i. Then it shows how to solve quadratic equations with imaginary solutions by taking square roots and expressing the solutions in terms of i. Examples are also provided of finding the complex zeros of quadratic functions by setting the functions equal to 0 and factoring. The document emphasizes that complex solutions come in conjugate pairs, and gives examples of finding the complex conjugate of solutions.
The document describes the number line and how it assigns numbers to points on a line. It explains that 0 is assigned to the center or origin of the line. Positive numbers are assigned to points to the right of 0, and negative numbers to points on the left. The number line defines the relative size of numbers based on their position, with numbers to the right being greater than those to the left. Intervals on the number line, such as -1 < x < 3, represent all the numbers between and including the bounds.
The document describes the rectangular coordinate system. Each point in the plane can be located using an ordered pair (x, y) representing horizontal and vertical distances from the origin. The x-axis represents horizontal distance and the y-axis represents vertical distance. Changing the x-value moves a point right or left, and changing the y-value moves a point up or down. The four quadrants are defined by the positive and negative x and y axes. Reflecting a point across an axis results in another point with the same distances but opposite sign for the axis coordinate.
This document discusses algebraic expressions and how to work with them. It defines an expression as a symbol or combination of symbols that represents a value or relation. It explains how to write verbal and symbolic expressions, calculate the value of an expression by plugging in numbers, and factor expressions by finding the greatest common factor or grouping terms. Examples are provided for each concept to illustrate the process. Resources for further information on algebraic expressions are listed at the end.
The document discusses methods for graphing quadratic equations. It explains that the graphs of quadratic equations are called parabolas, which are symmetric around a center line with a highest or lowest point called the vertex. It provides an example of graphing the equation y = x^2 - 4x - 12 by first finding the vertex, then making a table of symmetric x and y values around the vertex and plotting the points. The document also describes how to find the x-intercepts and y-intercepts of a parabola and provides an alternate two-step method for graphing a parabola.
To subtract two linear expressions:
1. Arrange the like terms in column form with the subtrahend expression underneath the minuend.
2. Take the additive inverse of the subtrahend expression by changing the sign of each term.
3. Add the subtrahend expression with its inverse to the minuend expression.
4. Simplify the resulting expression.
The document discusses number systems and provides examples of different types of numbers. It begins by explaining how early humans counted items without a formal system of numbers. The key developments were the creation of numbers and the number zero, which allowed people to answer questions about quantities.
The document then reviews natural numbers, whole numbers, and integers. It introduces rational numbers as numbers that can be expressed as fractions. Rational numbers can be positive or negative. Any number that cannot be expressed as a rational number, such as the square root of 2, is considered irrational. Real numbers include all rational and irrational numbers.
The document discusses quadratic equations and their properties over multiple years from 2005 to 2008. It provides example quadratic equations and asks the reader to find their roots, determine if they have equal roots, express constants in terms of other variables, and solve the equations. Specifically, it asks the reader to:
1) Find the value of p given one root of an equation.
2) Solve example quadratic equations and express a constant h in terms of k.
3) Determine possible values of p if an equation has two equal roots.
4) Solve another example quadratic equation to three decimal places.
The answers provided are: p = 5, x = 1/3 or x = -2, p = 8 or p
Solution manual for a survey of mathematics with applications 8th edition by ...AliDan123
Link download full: Solution Manual for A Survey of Mathematics with Applications 8th Edition by Allen R. Angel, Christine D. Abbott and Dennis C. Runde
https://digitalcontentmarket.org/download/solution-manual-for-a-survey-of-mathematics-with-applications-8th-edition-by-angel-abbott-and-runde/
a survey of mathematics with applications 8th edition solution manual
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Mathematics is the study of quantity, structure, space, and change. Mathematicians seek patterns and establish truth through rigorous logic and deduction from chosen axioms and definitions. The document then provides examples of different types of sequences including geometric sequences defined by multiplying a number each time, arithmetic sequences defined by adding a number each time, and special sequences like triangular and square numbers generated from patterns. It also discusses how to identify if a sequence is arithmetic, geometric, or harmonic based on relationships between terms.
This document discusses how to graph factorable polynomials by identifying the roots and their orders from the polynomial expression, making a sign chart, sketching the graph around each root based on the order, and connecting the pieces to obtain the full graph. As an example, it identifies the roots x=0, x=-2, and x=3 of order 1, 2, and 2 respectively from the polynomial P(x)=-x(x+2)2(x-3)2, makes the sign chart, and sketches the graph around each root to ultimately connect them into the full graph of P(x).
About sets , definition example, and some types of set. Explained the some operation of set like union of set and intersection of set with usual number example
The document contains questions and answers related to mathematics for senior high school. It includes questions from past national exams from 2000-2020, as well as sample questions in both the old and new testing systems. The questions cover topics like functions, limits, derivatives, and graphing. The document is authored by a mathematics teacher and intended as a review guide for students.
1. The document provides 10 problems involving quadratic equations. It gives the solutions to finding the range of values for parameters in quadratic equations so that the equations have certain properties like real roots or intersecting lines.
2. The solutions involve setting quadratic equations equal to zero and using the quadratic formula or factoring to solve for roots and parameters. Inequalities are used to determine ranges of parameter values.
3. The key steps shown are setting up and solving quadratic equations, using the quadratic formula or factoring, and determining parameter ranges through inequalities derived from having real or intersecting roots.
The document describes the rectangular coordinate system. Each point in a plane can be located using an ordered pair (x,y) where x represents the distance right or left from the origin and y represents the distance up or down. Changing the x-value moves the point right or left, and changing the y-value moves the point up or down. The plane is divided into four quadrants based on the sign of the x and y values. Reflecting a point across an axis results in another point with the same magnitude but opposite sign for the corresponding coordinate.
This document discusses different number systems including binary, hexadecimal, and octal. It begins by reviewing place value in the decimal system. It then explains how to convert between binary and decimal, including using addition tables to add binary numbers. Next, it covers the hexadecimal system and how to convert between hexadecimal and decimal. Finally, it introduces relations and equivalence relations, including properties of relations, partitions, and representing relations using matrices.
This document provides examples and explanations for performing operations with complex numbers. It begins with graphing complex numbers on a complex plane. It then covers determining the absolute value of complex numbers, adding and subtracting complex numbers by combining real and imaginary parts, and multiplying complex numbers using distribution and the property that i^2 = -1. It also discusses evaluating powers of i by expressing them as one of four values: i, -1, -i, or 1. The document aims to teach students how to perform basic operations like addition, subtraction, multiplication, and evaluating powers with complex numbers.
The document discusses factorable polynomials and graphing them. It defines a factorable polynomial P(x) as one that can be written as the product of linear factors P(x) = an(x - r1)(x - r2)...(x - rk), where r1, r2, etc. are the roots of P(x). It explains that for large values of |x|, the leading term of P(x) dominates so the graph resembles that of the leading term, while near the roots other terms contribute to the shape of the graph. Examples of graphs of polynomials like x^n are provided to illustrate the approach.
This document appears to be part of a Greek mathematics textbook. It contains definitions of common mathematical terms like function, graphical representation of a function, equality of functions, operations on functions, and composition of functions. It also defines what it means for a function to be increasing or decreasing over an interval of its domain. The document is divided into numbered sections and contains examples to illustrate each definition.
1) The document is an algebra II study guide containing 50 problems covering various algebra topics including properties, evaluating expressions, solving equations and inequalities, graphing lines and parabolas, working with complex numbers, and other concepts.
2) Many of the problems ask students to reference the specific section of their textbook that covers the relevant concept. The responses provide the answer and cite the textbook reference.
3) The study guide provides practice with essential algebra skills and will help students prepare for an exam or further study of advanced algebra topics.
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.
The document describes the rectangular coordinate system. Each point in the plane can be located using an ordered pair (x, y) representing horizontal and vertical distances from the origin. The x-axis represents horizontal distance and the y-axis represents vertical distance. Changing the x-value moves a point right or left, and changing the y-value moves a point up or down. The four quadrants are defined by the positive and negative x and y axes. Reflecting a point across an axis results in another point with the same distances but opposite sign for the axis coordinate.
This document discusses algebraic expressions and how to work with them. It defines an expression as a symbol or combination of symbols that represents a value or relation. It explains how to write verbal and symbolic expressions, calculate the value of an expression by plugging in numbers, and factor expressions by finding the greatest common factor or grouping terms. Examples are provided for each concept to illustrate the process. Resources for further information on algebraic expressions are listed at the end.
The document discusses methods for graphing quadratic equations. It explains that the graphs of quadratic equations are called parabolas, which are symmetric around a center line with a highest or lowest point called the vertex. It provides an example of graphing the equation y = x^2 - 4x - 12 by first finding the vertex, then making a table of symmetric x and y values around the vertex and plotting the points. The document also describes how to find the x-intercepts and y-intercepts of a parabola and provides an alternate two-step method for graphing a parabola.
To subtract two linear expressions:
1. Arrange the like terms in column form with the subtrahend expression underneath the minuend.
2. Take the additive inverse of the subtrahend expression by changing the sign of each term.
3. Add the subtrahend expression with its inverse to the minuend expression.
4. Simplify the resulting expression.
The document discusses number systems and provides examples of different types of numbers. It begins by explaining how early humans counted items without a formal system of numbers. The key developments were the creation of numbers and the number zero, which allowed people to answer questions about quantities.
The document then reviews natural numbers, whole numbers, and integers. It introduces rational numbers as numbers that can be expressed as fractions. Rational numbers can be positive or negative. Any number that cannot be expressed as a rational number, such as the square root of 2, is considered irrational. Real numbers include all rational and irrational numbers.
The document discusses quadratic equations and their properties over multiple years from 2005 to 2008. It provides example quadratic equations and asks the reader to find their roots, determine if they have equal roots, express constants in terms of other variables, and solve the equations. Specifically, it asks the reader to:
1) Find the value of p given one root of an equation.
2) Solve example quadratic equations and express a constant h in terms of k.
3) Determine possible values of p if an equation has two equal roots.
4) Solve another example quadratic equation to three decimal places.
The answers provided are: p = 5, x = 1/3 or x = -2, p = 8 or p
Solution manual for a survey of mathematics with applications 8th edition by ...AliDan123
Link download full: Solution Manual for A Survey of Mathematics with Applications 8th Edition by Allen R. Angel, Christine D. Abbott and Dennis C. Runde
https://digitalcontentmarket.org/download/solution-manual-for-a-survey-of-mathematics-with-applications-8th-edition-by-angel-abbott-and-runde/
a survey of mathematics with applications 8th edition solution manual
a survey of mathematics with applications pdf solution manual
solution manual a survey of mathematics with applications 8th edition pdf
a survey of mathematics with applications download solution manual
solution manual a survey of mathematics with applications free pdf
Mathematics is the study of quantity, structure, space, and change. Mathematicians seek patterns and establish truth through rigorous logic and deduction from chosen axioms and definitions. The document then provides examples of different types of sequences including geometric sequences defined by multiplying a number each time, arithmetic sequences defined by adding a number each time, and special sequences like triangular and square numbers generated from patterns. It also discusses how to identify if a sequence is arithmetic, geometric, or harmonic based on relationships between terms.
This document discusses how to graph factorable polynomials by identifying the roots and their orders from the polynomial expression, making a sign chart, sketching the graph around each root based on the order, and connecting the pieces to obtain the full graph. As an example, it identifies the roots x=0, x=-2, and x=3 of order 1, 2, and 2 respectively from the polynomial P(x)=-x(x+2)2(x-3)2, makes the sign chart, and sketches the graph around each root to ultimately connect them into the full graph of P(x).
About sets , definition example, and some types of set. Explained the some operation of set like union of set and intersection of set with usual number example
The document contains questions and answers related to mathematics for senior high school. It includes questions from past national exams from 2000-2020, as well as sample questions in both the old and new testing systems. The questions cover topics like functions, limits, derivatives, and graphing. The document is authored by a mathematics teacher and intended as a review guide for students.
1. The document provides 10 problems involving quadratic equations. It gives the solutions to finding the range of values for parameters in quadratic equations so that the equations have certain properties like real roots or intersecting lines.
2. The solutions involve setting quadratic equations equal to zero and using the quadratic formula or factoring to solve for roots and parameters. Inequalities are used to determine ranges of parameter values.
3. The key steps shown are setting up and solving quadratic equations, using the quadratic formula or factoring, and determining parameter ranges through inequalities derived from having real or intersecting roots.
The document describes the rectangular coordinate system. Each point in a plane can be located using an ordered pair (x,y) where x represents the distance right or left from the origin and y represents the distance up or down. Changing the x-value moves the point right or left, and changing the y-value moves the point up or down. The plane is divided into four quadrants based on the sign of the x and y values. Reflecting a point across an axis results in another point with the same magnitude but opposite sign for the corresponding coordinate.
This document discusses different number systems including binary, hexadecimal, and octal. It begins by reviewing place value in the decimal system. It then explains how to convert between binary and decimal, including using addition tables to add binary numbers. Next, it covers the hexadecimal system and how to convert between hexadecimal and decimal. Finally, it introduces relations and equivalence relations, including properties of relations, partitions, and representing relations using matrices.
This document provides examples and explanations for performing operations with complex numbers. It begins with graphing complex numbers on a complex plane. It then covers determining the absolute value of complex numbers, adding and subtracting complex numbers by combining real and imaginary parts, and multiplying complex numbers using distribution and the property that i^2 = -1. It also discusses evaluating powers of i by expressing them as one of four values: i, -1, -i, or 1. The document aims to teach students how to perform basic operations like addition, subtraction, multiplication, and evaluating powers with complex numbers.
The document discusses factorable polynomials and graphing them. It defines a factorable polynomial P(x) as one that can be written as the product of linear factors P(x) = an(x - r1)(x - r2)...(x - rk), where r1, r2, etc. are the roots of P(x). It explains that for large values of |x|, the leading term of P(x) dominates so the graph resembles that of the leading term, while near the roots other terms contribute to the shape of the graph. Examples of graphs of polynomials like x^n are provided to illustrate the approach.
This document appears to be part of a Greek mathematics textbook. It contains definitions of common mathematical terms like function, graphical representation of a function, equality of functions, operations on functions, and composition of functions. It also defines what it means for a function to be increasing or decreasing over an interval of its domain. The document is divided into numbered sections and contains examples to illustrate each definition.
1) The document is an algebra II study guide containing 50 problems covering various algebra topics including properties, evaluating expressions, solving equations and inequalities, graphing lines and parabolas, working with complex numbers, and other concepts.
2) Many of the problems ask students to reference the specific section of their textbook that covers the relevant concept. The responses provide the answer and cite the textbook reference.
3) The study guide provides practice with essential algebra skills and will help students prepare for an exam or further study of advanced algebra topics.
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.
CBSE Class 10 Mathematics Set theory Topic
Set theory Topics discussed in this document:
Introduction
Sets and their representations
Two methods of representing a set:
Roster form
Set-builder form
The empty set
Finite and infinite sets
Consider some examples
Equal sets
Subsets
Subsets of set of real numbers:
Intervals as subsets of R
Power set
Universal Set
Venn Diagrams
More Topics under Class 10th Set theory (CBSE):
Introduction to sets
Operations on sets
Visit Edvie.com for more topics
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 introduces the concept of sets. It defines a set as a well-defined collection of distinct objects, called elements or members. There are two main ways to represent a set: a roster form that lists the elements between curly brackets, and a set-builder form that uses a description to define the set of elements that satisfy a given property. The document discusses different types of sets such as empty, singleton, finite, and infinite sets. It also introduces set operations and relations like equivalent sets, equal sets, and subsets.
The document defines key concepts in set theory including:
1. A set is a well-defined collection of distinct objects called elements. Georg Cantor is credited with creating set theory.
2. Sets can be represented in roster form by listing elements within curly brackets or in set-builder form using properties the elements share.
3. Operations on sets include intersection, union, difference, and complement. Intersection is the set of common elements, union is all elements in either set, difference is elements only in the first set, and complement is elements not in the set.
- A set is a well-defined collection of objects. The empty set is a set with no elements. A set is finite if it has a definite number of elements, otherwise it is infinite.
- Two sets are equal if they have exactly the same elements. A set A is a subset of set B if every element of A is also an element of B.
- The power set of a set A is the collection of all subsets of A, including the empty set and A itself. It is denoted by P(A).
The document defines different types of sets and methods of representing sets. It discusses empty sets, singleton sets, finite and infinite sets. It also defines equivalent sets as sets with the same number of elements, and equal sets as sets containing the same elements. Disjoint sets are defined as sets that do not share any common elements. Examples are provided to illustrate these key set concepts and relationships between sets.
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.
This document provides information about sets, relations, and functions in mathematics. It begins by giving examples of sets and non-sets to illustrate what makes a collection a well-defined set. It then defines various set concepts like finite and infinite sets, the empty set, singleton sets, equal sets, subsets, unions and intersections of sets. It introduces the concept of relations and functions, defining a function as a special type of relation. It concludes by stating the objectives of learning about sets, relations and functions.
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, such as using capital letters to represent sets and lowercase letters for elements. Two methods for representing sets are described: roster form and set-builder form. The document then classifies sets as finite or infinite, empty, singleton, equal, equivalent, and disjoint sets. Examples are provided to illustrate each concept.
Moazzzim Sir (25.07.23)CSE 1201, Week#3, Lecture#7.pptxKhalidSyfullah6
This document provides an overview of key concepts in set theory including:
- The definition of a set as an unordered collection of distinct elements
- Common ways to describe and represent sets such as listing elements, set-builder notation, and Venn diagrams
- Important set terminology including subset, proper subset, set equality, cardinality (size of a set), finite vs infinite sets, power set, and Cartesian product
The document uses examples and explanations to illustrate each concept over 34 pages. It appears to be lecture material introducing students to the basic foundations of set theory.
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.
This document provides information about visually representing sets using different methods. It discusses listing sets using curly brackets, representing sets using set-builder notation with conditions, and displaying sets using Venn diagrams. The document also covers labeling sets, finding cardinality, types of sets such as empty, finite, infinite, and singleton sets, and relationships between sets like equal, equivalent, subset, and superset.
This document provides an introduction to the concept of sets in mathematics. It defines what a set is and explains that a set is a well-defined collection of distinct objects. It discusses different ways of representing sets, such as using a roster or set-builder notation. It also covers various types of sets like finite, infinite, empty, singleton, equivalent and equal sets. The document explains set operations like union, intersection and complement. It introduces Venn diagrams to visually represent relationships between sets. Finally, it discusses algebraic properties that set operations satisfy, such as commutative, associative and distributive laws.
Rd sharma-class-11-maths-chapter-1-sets (1)Sarravanan R
This document provides solutions to exercises from Chapter 1 on Sets from the RD Sharma Solutions for Class 11 Maths textbook. The first exercise defines the difference between a collection and a set and identifies which of several examples are sets based on whether they are well-defined. The second exercise involves describing sets in roster form and set-builder form. The third exercise lists the elements of various sets defined in set-builder form.
The document discusses integers and their properties. It defines integers as numbers that can be written without fractions or decimals, such as 21, 4, and -2048, but not 9.75 or √2. Integers include natural numbers, zero, and their negatives. They form the smallest group containing natural numbers under addition. Integers also form a ring with unique homomorphisms to other rings, characterizing their fundamental nature.
The document contains a pie chart showing the percentage of students enrolled in different hobby classes at a school with a total student population of 3600. It then provides 5 questions and solutions related to calculating percentages and numbers of students enrolled in different classes based on the data in the pie chart. The questions include calculating percentages of students in one class compared to another, determining the number of students enrolled in a specific class, finding percentages of one class compared to another, determining ratios of enrollment between classes, and calculating total enrollment between two classes.
Simplification and data interpretation tableSurya Swaroop
This document contains a table with the number of books sold (in thousands) by different shops (P, Q, R, S, T) in different years (2001-2005). It also contains 9 questions that require referencing data from the table to solve. The questions include calculating ratios of books sold between shops in a given year, identifying the shop with the maximum total sales across all years, and calculating differences and totals of books sold by shops in specified years.
Environmental engineering textbook - civil engineering
topics-
Quality and Analysis of Water
Treatment of Water
Disinfection
Distribution of Water
Quality and Analysis of Water
water demand and Quantity Estimation
Sources of Water
Collection and Conveyance of Water etc .... more
engineering drawing - Development, METHOD OF Development, development of surfaces of solids, Development of a Right regular cylinder, Development of a Right Rectangular Prism, Truncated Cylinder.
*elasticity, elastic limit, stress, strain, and ultimate strength.
*Write and apply formulas for calculating Young’s modulus, shear modulus, and bulk modulus.
*types of stress
*longitudinal stress and strain
*elastic limit
*modulus of elasticity
*Solve problems involving each of the parameters in the above objectives.
The document is a scanned brochure that provides information about JNTU Fast updates, a website that offers updates and notifications about academic events, timetables, results, and other announcements related to Jawaharlal Nehru Technological University. The brochure encourages students to visit the website or sign up for alerts to get important updates from their university.
Design of Columns: Built-up compression members – Design of lacings and battens. Design Principles of Eccentrically loaded columns, Splicing of columns.
INTRODUCTION: Advantages and Disadvantages of Steel structures, Loads and Load
combinations, Design considerations, Limit State Method (LSM) of design, Failure criteria for
steel, Codes, Specifications and section classification.
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maths
1. Chapter 1
SETS
Georg Cantor
(1845-1918)
In these days of conflict between ancient and modern studies; there
must surely be something to be said for a study which did not
begin with Pythagoras and will not end with Einstein; but
is the oldest and the youngest. — G.H. HARDY
1.1 Introduction
The concept of set serves as a fundamental part of the
present day mathematics. Today this concept is being used
in almost every branch of mathematics. Sets are used to
define the concepts of relations and functions. The study of
geometry, sequences, probability, etc. requires the knowledge
of sets.
The theory of sets was developed by German
mathematician Georg Cantor (1845-1918). He first
encounteredsetswhileworkingon“problemsontrigonometric
series”. In this Chapter, we discuss some basic definitions
and operations involving sets.
1.2 Sets and their Representations
In everyday life, we often speak of collections of objects of a particular kind, such as,
a pack of cards, a crowd of people, a cricket team, etc. In mathematics also, we come
across collections, for example, of natural numbers, points, prime numbers, etc. More
specially, we examine the following collections:
(i) Odd natural numbers less than 10, i.e., 1, 3, 5, 7, 9
(ii) The rivers of India
(iii) The vowels in the English alphabet, namely, a, e, i, o, u
(iv) Various kinds of triangles
(v) Prime factors of 210, namely, 2,3,5 and 7
(vi) The solution of the equation: x2
– 5x + 6 = 0, viz, 2 and 3.
We note that each of the above example is a well-defined collection of objects in
2018-19
2. 2 MATHEMATICS
the sense that we can definitely decide whether a given particular object belongs to a
given collection or not. For example, we can say that the river Nile does not belong to
the collection of rivers of India. On the other hand, the river Ganga does belong to this
colleciton.
We give below a few more examples of sets used particularly in mathematics, viz.
N : the set of all natural numbers
Z : the set of all integers
Q : the set of all rational numbers
R : the set of real numbers
Z+
: the set of positive integers
Q+
: the set of positive rational numbers, and
R+
: the set of positive real numbers.
The symbols for the special sets given above will be referred to throughout
this text.
Again the collection of five most renowned mathematicians of the world is not
well-defined, because the criterion for determining a mathematician as most renowned
may vary from person to person. Thus, it is not a well-defined collection.
We shall say that a set is a well-defined collection of objects.
The following points may be noted :
(i) Objects, elements and members of a set are synonymous terms.
(ii) Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc.
(iii) The elements of a set are represented by small letters a, b, c, x, y, z, etc.
If a is an element of a set A, we say that “ a belongs to A” the Greek symbol ∈
(epsilon) is used to denote the phrase ‘belongs to’. Thus, we write a ∈ A. If ‘b’ is not
an element of a set A, we write b ∉ A and read “b does not belong to A”.
Thus, in the set V of vowels in the English alphabet, a ∈ V but b ∉ V. In the set
P of prime factors of 30, 3 ∈ P but 15 ∉ P.
There are two methods of representing a set :
(i) Roster or tabular form
(ii) Set-builder form.
(i) In roster form, all the elements of a set are listed, the elements are being separated
by commas and are enclosed within braces { }. For example, the set of all even
positive integers less than 7 is described in roster form as {2, 4, 6}. Some more
examples of representing a set in roster form are given below :
(a) The set of all natural numbers which divide 42 is {1, 2, 3, 6, 7, 14, 21, 42}.
2018-19
3. SETS 3
Note In roster form, the order in which the elements are listed is immaterial.
Thus, the above set can also be represented as {1, 3, 7, 21, 2, 6, 14, 42}.
(b) The set of all vowels in the English alphabet is {a, e, i, o, u}.
(c) The set of odd natural numbers is represented by {1, 3, 5, . . .}. The dots
tell us that the list of odd numbers continue indefinitely.
Note It may be noted that while writing the set in roster form an element is not
generally repeated, i.e., all the elements are taken as distinct. For example, the set
of letters forming the word ‘SCHOOL’ is { S, C, H, O, L} or {H, O, L, C, S}. Here,
the order of listing elements has no relevance.
(ii) In set-builder form, all the elements of a set possess a single common property
which is not possessed by any element outside the set. For example, in the set
{a, e, i, o, u}, all the elements possess a common property, namely, each of them
is a vowel in the English alphabet, and no other letter possess this property. Denoting
this set by V, we write
V = {x : x is a vowel in English alphabet}
It may be observed that we describe the element of the set by using a symbol x
(any other symbol like the letters y, z, etc. could be used) which is followed by a colon
“ : ”. After the sign of colon, we write the characteristic property possessed by the
elements of the set and then enclose the whole description within braces. The above
description of the set V is read as “the set of all x such that x is a vowel of the English
alphabet”. In this description the braces stand for “the set of all”, the colon stands for
“such that”. For example, the set
A = {x : x is a natural number and 3 < x < 10} is read as “the set of all x such that
x is a natural number and x lies between 3 and 10.” Hence, the numbers 4, 5, 6,
7, 8 and 9 are the elements of the set A.
If we denote the sets described in (a), (b) and (c) above in roster form by A, B,
C, respectively, then A, B, C can also be represented in set-builder form as follows:
A= {x : x is a natural number which divides 42}
B= {y : y is a vowel in the English alphabet}
C= {z : z is an odd natural number}
Example 1 Write the solution set of the equation x2
+ x – 2 = 0 in roster form.
Solution The given equation can be written as
(x – 1) (x + 2) = 0, i. e., x = 1, – 2
Therefore, the solution set of the given equation can be written in roster form as {1, – 2}.
Example 2 Write the set {x : x is a positive integer and x2
< 40} in the roster form.
2018-19
4. 4 MATHEMATICS
Solution The required numbers are 1, 2, 3, 4, 5, 6. So, the given set in the roster form
is {1, 2, 3, 4, 5, 6}.
Example 3 Write the set A = {1, 4, 9, 16, 25, . . . }in set-builder form.
Solution We may write the set A as
A = {x : x is the square of a natural number}
Alternatively, we can write
A = {x : x = n2
, where n ∈ N}
Example 4 Write the set
1 2 3 4 5 6
{ }
2 3 4 5 6 7
, , , , , in the set-builder form.
Solution We see that each member in the given set has the numerator one less than
the denominator. Also, the numerator begin from 1 and do not exceed 6. Hence, in the
set-builder form the given set is
where is a natural number and 1 6
1
n
x : x , n n
n
= ≤ ≤
+
Example 5 Match each of the set on the left described in the roster form with the
same set on the right described in the set-builder form :
(i) {P, R, I, N, C, A, L} (a) { x : x is a positive integer and is a divisor of 18}
(ii) { 0 } (b) { x : x is an integer and x2
– 9 = 0}
(iii) {1, 2, 3, 6, 9, 18} (c) {x : x is an integer and x + 1= 1}
(iv) {3, –3} (d) {x : x is a letter of the word PRINCIPAL}
Solution Since in (d), there are 9 letters in the word PRINCIPAL and two letters P and I
are repeated, so (i) matches (d). Similarly, (ii) matches (c) as x + 1 = 1 implies
x = 0.Also, 1, 2 ,3, 6, 9, 18 are all divisors of 18 and so (iii) matches (a). Finally, x2
– 9 = 0
implies x = 3, –3 and so (iv) matches (b).
EXERCISE 1.1
1. Which of the following are sets ? Justify your answer.
(i) The collection of all the months of a year beginning with the letter J.
(ii) The collection of ten most talented writers of India.
(iii) A team of eleven best-cricket batsmen of the world.
(iv) The collection of all boys in your class.
(v) The collection of all natural numbers less than 100.
(vi) A collection of novels written by the writer Munshi Prem Chand.
(vii) The collection of all even integers.
2018-19
5. SETS 5
(viii) The collection of questions in this Chapter.
(ix) A collection of most dangerous animals of the world.
2. Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank
spaces:
(i) 5. . .A (ii) 8 . . . A (iii) 0. . .A
(iv) 4. . . A (v) 2. . .A (vi) 10. . .A
3. Write the following sets in roster form:
(i) A = {x : x is an integer and –3 ≤ x < 7}
(ii) B = {x : x is a natural number less than 6}
(iii) C = {x : x is a two-digit natural number such that the sum of its digits is 8}
(iv) D = {x : x is a prime number which is divisor of 60}
(v) E = The set of all letters in the word TRIGONOMETRY
(vi) F = The set of all letters in the word BETTER
4. Write the following sets in the set-builder form :
(i) (3, 6, 9, 12} (ii) {2,4,8,16,32} (iii) {5, 25, 125, 625}
(iv) {2, 4, 6, . . .} (v) {1,4,9, . . .,100}
5. List all the elements of the following sets :
(i) A = {x : x is an odd natural number}
(ii) B = {x : x is an integer,
1
2
– < x <
9
2
}
(iii) C = {x : x is an integer, x2
≤ 4}
(iv) D = {x : x is a letter in the word “LOYAL”}
(v) E = {x : x is a month of a year not having 31 days}
(vi) F = {x : x is a consonant in the English alphabet which precedes k }.
6. Match each of the set on the left in the roster form with the same set on the right
described in set-builder form:
(i) {1, 2, 3, 6} (a) {x : x is a prime number and a divisor of 6}
(ii) {2, 3} (b) {x : x is an odd natural number less than 10}
(iii) {M,A,T,H,E,I,C,S} (c) {x : x is natural number and divisor of 6}
(iv) {1, 3, 5, 7, 9} (d) {x : x is a letter of the word MATHEMATICS}.
1.3 The Empty Set
Consider the set
A = { x : x is a student of Class XI presently studying in a school }
We can go to the school and count the number of students presently studying in
Class XI in the school. Thus, the set A contains a finite number of elements.
We now write another set B as follows:
2018-19
6. 6 MATHEMATICS
B = { x : x is a student presently studying in both Classes X and XI }
We observe that a student cannot study simultaneously in both Classes X and XI.
Thus, the set B contains no element at all.
Definition 1 A set which does not contain any element is called the empty set or the
null set or the void set.
According to this definition, B is an empty set while A is not an empty set. The
empty set is denoted by the symbol φ or { }.
We give below a few examples of empty sets.
(i) Let A = {x : 1 < x < 2, x is a natural number}. Then A is the empty set,
because there is no natural number between 1 and 2.
(ii) B = {x : x2
– 2 = 0 and x is rational number}. Then B is the empty set because
the equation x2
– 2 = 0 is not satisfied by any rational value of x.
(iii) C = {x : x is an even prime number greater than 2}.Then C is the empty set,
because 2 is the only even prime number.
(iv) D = { x : x2
= 4, x is odd }. Then D is the empty set, because the equation
x2
= 4 is not satisfied by any odd value of x.
1.4 Finite and Infinite Sets
Let A = {1, 2, 3, 4, 5}, B = {a, b, c, d, e, g}
and C = { men living presently in different parts of the world}
We observe thatAcontains 5 elements and B contains 6 elements. How many elements
does C contain? As it is, we do not know the number of elements in C, but it is some
natural number which may be quite a big number. By number of elements of a set S,
we mean the number of distinct elements of the set and we denote it by n (S). If n (S)
is a natural number, then S is non-empty finite set.
Consider the set of natural numbers. We see that the number of elements of this
set is not finite since there are infinite number of natural numbers. We say that the set
of natural numbers is an infinite set. The sets A, B and C given above are finite sets
and n(A) = 5, n(B) = 6 and n(C) = some finite number.
Definition 2 A set which is empty or consists of a definite number of elements is
called finite otherwise, the set is called infinite.
Consider some examples :
(i) Let W be the set of the days of the week. Then W is finite.
(ii) Let S be the set of solutions of the equation x2
–16 = 0. Then S is finite.
(iii) Let G be the set of points on a line. Then G is infinite.
When we represent a set in the roster form, we write all the elements of the set
within braces { }. It is not possible to write all the elements of an infinite set within
braces { } because the numbers of elements of such a set is not finite. So, we represent
2018-19
7. SETS 7
some infinite set in the roster form by writing a few elements which clearly indicate the
structure of the set followed ( or preceded ) by three dots.
For example, {1, 2, 3 . . .} is the set of natural numbers, {1, 3, 5, 7, . . .} is the set
of odd natural numbers, {. . .,–3, –2, –1, 0,1, 2 ,3, . . .} is the set of integers. All these
sets are infinite.
Note All infinite sets cannot be described in the roster form. For example, the
set of real numbers cannot be described in this form, because the elements of this
set do not follow any particular pattern.
Example 6 State which of the following sets are finite or infinite :
(i) {x : x ∈ N and (x – 1) (x –2) = 0}
(ii) {x : x ∈ N and x2
= 4}
(iii) {x : x ∈ N and 2x –1 = 0}
(iv) {x : x ∈ N and x is prime}
(v) {x : x ∈ N and x is odd}
Solution (i) Given set = {1, 2}. Hence, it is finite.
(ii) Given set = {2}. Hence, it is finite.
(iii) Given set = φ. Hence, it is finite.
(iv) The given set is the set of all prime numbers and since set of prime
numbers is infinite. Hence the given set is infinite
(v) Since there are infinite number of odd numbers, hence, the given set is
infinite.
1.5 Equal Sets
Given two sets A and B, if every element of A is also an element of B and if every
element of B is also an element of A, then the sets A and B are said to be equal.
Clearly, the two sets have exactly the same elements.
Definition 3 Two sets A and B are said to be equal if they have exactly the same
elements and we write A = B. Otherwise, the sets are said to be unequal and we write
A ≠ B.
We consider the following examples :
(i) Let A = {1, 2, 3, 4} and B = {3, 1, 4, 2}. Then A = B.
(ii) Let A be the set of prime numbers less than 6 and P the set of prime factors
of 30. ThenA and P are equal, since 2, 3 and 5 are the only prime factors of
30 and also these are less than 6.
Note A set does not change if one or more elements of the set are repeated.
For example, the sets A = {1, 2, 3} and B = {2, 2, 1, 3, 3} are equal, since each
2018-19
8. 8 MATHEMATICS
element of A is in B and vice-versa. That is why we generally do not repeat any
element in describing a set.
Example 7 Find the pairs of equal sets, if any, give reasons:
A = {0}, B = {x : x > 15 and x < 5},
C = {x : x – 5 = 0 }, D = {x: x2
= 25},
E = {x : x is an integral positive root of the equation x2
– 2x –15 = 0}.
Solution Since 0 ∈ A and 0 does not belong to any of the sets B, C, D and E, it
follows that, A ≠ B, A ≠ C, A ≠ D, A ≠ E.
Since B = φ but none of the other sets are empty. Therefore B ≠ C, B ≠ D
and B ≠ E. Also C = {5} but –5 ∈ D, hence C ≠ D.
Since E = {5}, C = E. Further, D = {–5, 5} and E = {5}, we find that, D ≠ E.
Thus, the only pair of equal sets is C and E.
Example 8 Which of the following pairs of sets are equal? Justify your answer.
(i) X, the set of letters in “ALLOY” and B, the set of letters in “LOYAL”.
(ii) A = {n : n ∈ Z and n2
≤ 4} and B = {x : x ∈ R and x2
– 3x + 2 = 0}.
Solution (i) We have, X = {A, L, L, O, Y}, B = {L, O, Y, A, L}. Then X and B are
equal sets as repetition of elements in a set do not change a set. Thus,
X = {A, L, O, Y} = B
(ii)A= {–2, –1, 0, 1, 2}, B = {1, 2}. Since 0 ∈Aand 0 ∉ B, Aand B are not equal sets.
EXERCISE 1.2
1. Which of the following are examples of the null set
(i) Set of odd natural numbers divisible by 2
(ii) Set of even prime numbers
(iii) { x : x is a natural numbers, x < 5 and x > 7 }
(iv) { y : y is a point common to any two parallel lines}
2. Which of the following sets are finite or infinite
(i) The set of months of a year
(ii) {1, 2, 3, . . .}
(iii) {1, 2, 3, . . .99, 100}
(iv) The set of positive integers greater than 100
(v) The set of prime numbers less than 99
3. State whether each of the following set is finite or infinite:
(i) The set of lines which are parallel to the x-axis
(ii) The set of letters in the English alphabet
(iii) The set of numbers which are multiple of 5
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9. SETS 9
(iv) The set of animals living on the earth
(v) The set of circles passing through the origin (0,0)
4. In the following, state whether A = B or not:
(i) A = { a, b, c, d } B = { d, c, b, a }
(ii) A = { 4, 8, 12, 16 } B = { 8, 4, 16, 18}
(iii) A = {2, 4, 6, 8, 10} B = { x : x is positive even integer and x ≤ 10}
(iv) A = { x : x is a multiple of 10}, B = { 10, 15, 20, 25, 30, . . . }
5. Are the following pair of sets equal ? Give reasons.
(i) A = {2, 3}, B = {x : x is solution of x2
+ 5x + 6 = 0}
(ii) A = { x : x is a letter in the word FOLLOW}
B = { y : y is a letter in the word WOLF}
6. From the sets given below, select equal sets :
A = { 2, 4, 8, 12}, B = { 1, 2, 3, 4}, C = { 4, 8, 12, 14}, D = { 3, 1, 4, 2}
E = {–1, 1}, F = { 0, a}, G = {1, –1}, H = { 0, 1}
1.6 Subsets
Consider the sets : X = set of all students in your school, Y = set of all students in your
class.
We note that every element of Y is also an element of X; we say that Y is a subset
of X. The fact that Y is subset of X is expressed in symbols as Y ⊂ X. The symbol ⊂
stands for ‘is a subset of’ or ‘is contained in’.
Definition 4 A set A is said to be a subset of a set B if every element of A is also an
element of B.
In other words, A ⊂ B if whenever a ∈ A, then a ∈ B. It is often convenient to
use the symbol “⇒” which means implies. Using this symbol, we can write the definiton
of subset as follows:
A ⊂ B if a ∈ A ⇒ a ∈ B
We read the above statement as “A is a subset of B if a is an element of A
implies that a is also an element of B”. If A is not a subset of B, we write A ⊄ B.
We may note that for A to be a subset of B, all that is needed is that every
element of A is in B. It is possible that every element of B may or may not be in A. If
it so happens that every element of B is also inA, then we shall also have B ⊂A. In this
case, A and B are the same sets so that we have A ⊂ B and B ⊂ A ⇔ A = B, where
“⇔” is a symbol for two way implications, and is usually read as if and only if (briefly
written as “iff”).
It follows from the above definition that every set A is a subset of itself, i.e.,
A ⊂ A. Since the empty set φ has no elements, we agree to say that φ is a subset of
every set. We now consider some examples :
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10. 10 MATHEMATICS
(i) The set Q of rational numbers is a subset of the set R of real numbes, and
we write Q ⊂ R.
(ii) If A is the set of all divisors of 56 and B the set of all prime divisors of 56,
then B is a subset of A and we write B ⊂ A.
(iii) LetA = {1, 3, 5} and B = {x : x is an odd natural number less than 6}. Then
A ⊂ B and B ⊂ A and hence A = B.
(iv) Let A = { a, e, i, o, u} and B = { a, b, c, d}. Then A is not a subset of B,
also B is not a subset of A.
Let A and B be two sets. If A ⊂ B and A ≠ B , then A is called a proper subset
of B and B is called superset of A. For example,
A = {1, 2, 3} is a proper subset of B = {1, 2, 3, 4}.
If a set A has only one element, we call it a singleton set. Thus,{ a } is a
singleton set.
Example 9 Consider the sets
φ, A = { 1, 3 }, B = {1, 5, 9}, C = {1, 3, 5, 7, 9}.
Insert the symbol ⊂ or ⊄ between each of the following pair of sets:
(i) φ . . . B (ii) A . . . B (iii) A . . . C (iv) B . . . C
Solution (i) φ ⊂ B as φ is a subset of every set.
(ii) A ⊄ B as 3 ∈ A and 3 ∉ B
(iii) A ⊂ C as 1, 3 ∈ A also belongs to C
(iv) B ⊂ C as each element of B is also an element of C.
Example 10 Let A = { a, e, i, o, u} and B = { a, b, c, d}. Is A a subset of B ? No.
(Why?). Is B a subset of A? No. (Why?)
Example 11 Let A, B and C be three sets. If A ∈ B and B ⊂ C, is it true that
A ⊂ C?. If not, give an example.
Solution No. Let A = {1}, B = {{1}, 2} and C = {{1}, 2, 3}. Here A ∈ B as A = {1}
and B ⊂ C. But A ⊄ C as 1 ∈ A and 1 ∉ C.
Note that an element of a set can never be a subset of itself.
1.6.1 Subsets of set of real numbers
As noted in Section 1.6, there are many important subsets of R. We give below the
names of some of these subsets.
The set of natural numbers N = {1, 2, 3, 4, 5, . . .}
The set of integers Z = {. . ., –3, –2, –1, 0, 1, 2, 3, . . .}
The set of rational numbers Q = { x : x =
p
q
, p, q ∈ Z and q ≠ 0}
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11. SETS 11
which is read “ Q is the set of all numbers x such that x equals the quotient
p
q , where
p and q are integers and q is not zero”. Members of Q include –5 (which can be
expressed as
5
1
– ) ,
7
5
,
1
3
2
(which can be expressed as
7
2
) and
11
3
– .
The set of irrational numbers, denoted by T, is composed of all other real numbers.
Thus T = {x : x ∈ R and x ∉ Q}, i.e., all real numbers that are not rational.
Members of T include 2 , 5 and π .
Some of the obvious relations among these subsets are:
N ⊂ Z ⊂ Q, Q ⊂ R, T ⊂ R, N ⊄ T.
1.6.2 Intervals as subsets of R Let a, b ∈ R and a < b. Then the set of real numbers
{ y : a < y < b} is called an open interval and is denoted by (a, b). All the points
between a and b belong to the open interval (a, b) but a, b themselves do not belong to
this interval.
The interval which contains the end points also is called closed interval and is
denoted by [ a, b ]. Thus
[ a, b ] = {x : a ≤ x ≤ b}
We can also have intervals closed at one end and open at the other, i.e.,
[ a, b ) = {x : a ≤ x < b} is an open interval from a to b, including a but excluding b.
( a, b ] = { x : a < x ≤ b } is an open interval from a to b including b but excluding a.
These notations provide an alternative way of designating the subsets of set of
real numbers. For example , if A = (–3, 5) and B = [–7, 9], then A ⊂ B. The set [ 0, ∞)
defines the set of non-negative real numbers, while set ( – ∞, 0 ) defines the set of
negative real numbers. The set ( – ∞, ∞ ) describes the set of real numbers in relation
to a line extending from – ∞ to ∞.
On real number line, various types of intervals described above as subsets of R,
are shown in the Fig 1.1.
Here, we note that an interval contains infinitely many points.
For example, the set {x : x ∈ R, –5 < x ≤ 7}, written in set-builder form, can be
written in the form of interval as (–5, 7] and the interval [–3, 5) can be written in set-
builder form as {x : –3 ≤ x < 5}.
Fig 1.1
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12. 12 MATHEMATICS
The number (b – a) is called the length of any of the intervals (a, b), [a, b],
[a, b) or (a, b].
1.7 Power Set
Consider the set {1, 2}. Let us write down all the subsets of the set {1, 2}. We
know that φ is a subset of every set . So, φ is a subset of {1, 2}. We see that {1}
and { 2 }are also subsets of {1, 2}. Also, we know that every set is a subset of
itself. So, { 1, 2 } is a subset of {1, 2}. Thus, the set { 1, 2 } has, in all, four
subsets, viz. φ, { 1 }, { 2 } and { 1, 2 }. The set of all these subsets is called the
power set of { 1, 2 }.
Definition 5 The collection of all subsets of a set A is called the power set of A. It is
denoted by P(A). In P(A), every element is a set.
Thus, as in above, if A = { 1, 2 }, then
P( A ) = { φ,{ 1 }, { 2 }, { 1,2 }}
Also, note that n [ P (A) ] = 4 = 22
In general, if A is a set with n(A) = m, then it can be shown that
n [ P(A)] = 2m
.
1.8 Universal Set
Usually, in a particular context, we have to deal with the elements and subsets of a
basic set which is relevant to that particular context. For example, while studying the
system of numbers, we are interested in the set of natural numbers and its subsets such
as the set of all prime numbers, the set of all even numbers, and so forth. This basic set
is called the “Universal Set”. The universal set is usually denoted by U, and all its
subsets by the letters A, B, C, etc.
For example, for the set of all integers, the universal set can be the set of rational
numbers or, for that matter, the set R of real numbers. For another example, in human
population studies, the universal set consists of all the people in the world.
EXERCISE 1.3
1. Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces :
(i) { 2, 3, 4 } . . . { 1, 2, 3, 4,5 } (ii) { a, b, c } . . . { b, c, d }
(iii) {x : x is a student of Class XI of your school}. . .{x : x student of your school}
(iv) {x : x is a circle in the plane} . . .{x : x is a circle in the same plane with
radius 1 unit}
(v) {x : x is a triangle in a plane} . . . {x : x is a rectangle in the plane}
(vi) {x:xisanequilateraltriangleinaplane}... {x:xisatriangleinthesameplane}
(vii) {x : x is an even natural number} . . . {x : x is an integer}
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13. SETS 13
Fig 1.2
2. Examine whether the following statements are true or false:
(i) { a, b } ⊄ { b, c, a }
(ii) { a, e } ⊂ { x : x is a vowel in the English alphabet}
(iii) { 1, 2, 3 } ⊂ { 1, 3, 5 }
(iv) { a } ⊂ { a, b, c }
(v) { a } ∈ { a, b, c }
(vi) { x : x is an even natural number less than 6} ⊂ { x : x is a natural number
which divides 36}
3. Let A= { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?
(i) {3, 4} ⊂ A (ii) {3, 4} ∈ A (iii) {{3, 4}} ⊂ A
(iv) 1 ∈ A (v) 1 ⊂ A (vi) {1, 2, 5} ⊂ A
(vii) {1, 2, 5} ∈ A (viii) {1, 2, 3} ⊂ A (ix) φ ∈ A
(x) φ ⊂ A (xi) {φ} ⊂ A
4. Write down all the subsets of the following sets
(i) {a} (ii) {a, b} (iii) {1, 2, 3} (iv) φ
5. How many elements has P(A), if A = φ?
6. Write the following as intervals :
(i) {x : x ∈ R, – 4 < x ≤ 6} (ii) {x : x ∈ R, – 12 < x < –10}
(iii) {x : x ∈ R, 0 ≤ x < 7} (iv) {x : x ∈ R, 3 ≤ x ≤ 4}
7. Write the following intervals in set-builder form :
(i) (– 3, 0) (ii) [6 , 12] (iii) (6, 12] (iv) [–23, 5)
8. What universal set(s) would you propose for each of the following :
(i) The set of right triangles. (ii) The set of isosceles triangles.
9. Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the
following may be considered as universal set (s) for all the three sets A, B and C
(i) {0, 1, 2, 3, 4, 5, 6}
(ii) φ
(iii) {0,1,2,3,4,5,6,7,8,9,10}
(iv) {1,2,3,4,5,6,7,8}
1.9 Venn Diagrams
Most of the relationships between sets can be
represented by means of diagrams which are known
as Venn diagrams. Venn diagrams are named after
the English logician, John Venn (1834-1883). These
diagrams consist of rectangles and closed curves
usually circles. The universal set is represented
usually by a rectangle and its subsets by circles.
In Venn diagrams, the elements of the sets
are written in their respective circles (Figs 1.2 and 1.3)
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14. 14 MATHEMATICS
Fig 1.3
Illustration 1 In Fig 1.2, U = {1,2,3, ..., 10} is the
universal set of which
A = {2,4,6,8,10} is a subset.
Illustration 2 In Fig 1.3, U = {1,2,3, ..., 10} is the
universal set of which
A = {2,4,6,8,10} and B = {4, 6} are subsets,
and also B ⊂ A.
The reader will see an extensive use of the
Venn diagrams when we discuss the union, intersection and difference of sets.
1.10 Operations on Sets
In earlier classes, we have learnt how to perform the operations of addition, subtraction,
multiplication and division on numbers. Each one of these operations was performed
on a pair of numbers to get another number. For example, when we perform the
operation of addition on the pair of numbers 5 and 13, we get the number 18. Again,
performing the operation of multiplication on the pair of numbers 5 and 13, we get 65.
Similarly, there are some operations which when performed on two sets give rise to
another set. We will now define certain operations on sets and examine their properties.
Henceforth, we will refer all our sets as subsets of some universal set.
1.10.1 Union of sets Let A and B be any two sets. The union of A and B is the set
which consists of all the elements ofAand all the elements of B, the common elements
being taken only once. The symbol ‘∪’ is used to denote the union. Symbolically, we
write A ∪ B and usually read as ‘A union B’.
Example 12 Let A = { 2, 4, 6, 8} and B = { 6, 8, 10, 12}. Find A ∪ B.
Solution We have A ∪ B = { 2, 4, 6, 8, 10, 12}
Note that the common elements 6 and 8 have been taken only once while writing
A ∪ B.
Example 13 Let A = { a, e, i, o, u } and B = { a, i, u }. Show that A ∪ B = A
Solution We have, A ∪ B = { a, e, i, o, u } = A.
This example illustrates that union of sets A and its subset B is the set A
itself, i.e., if B ⊂ A, then A ∪ B = A.
Example 14 Let X = {Ram, Geeta,Akbar} be the set of students of Class XI, who are
in school hockey team. Let Y = {Geeta, David, Ashok} be the set of students from
Class XI who are in the school football team. Find X ∪ Y and interpret the set.
Solution We have, X ∪ Y = {Ram, Geeta, Akbar, David, Ashok}. This is the set of
students from Class XI who are in the hockey team or the football team or both.
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15. SETS 15
Thus, we can define the union of two sets as follows:
Definition 6 The union of two sets A and B is the set C which consists of all those
elements which are either in A or in B (including
those which are in both). In symbols, we write.
A ∪ B = { x : x ∈A or x ∈B }
The union of two sets can be represented by a
Venn diagram as shown in Fig 1.4.
The shaded portion in Fig 1.4 representsA∪ B.
Some Properties of the Operation of Union
(i) A ∪ B = B ∪ A (Commutative law)
(ii) ( A ∪ B ) ∪ C = A ∪ ( B ∪ C)
(Associative law )
(iii) A ∪ φ = A (Law of identity element, φ is the identity of ∪)
(iv) A ∪ A = A (Idempotent law)
(v) U ∪ A = U (Law of U)
1.10.2 Intersection of sets The intersection of sets A and B is the set of all elements
which are common to bothAand B. The symbol ‘∩’is used to denote the intersection.
The intersection of two sets A and B is the set of all those elements which belong to
both A and B. Symbolically, we write A ∩ B = {x : x ∈ A and x ∈ B}.
Example 15 Consider the sets A and B of Example 12. Find A ∩ B.
Solution We see that 6, 8 are the only elements which are common to both A and B.
Hence A ∩ B = { 6, 8 }.
Example 16 Consider the sets X and Y of Example 14. Find X ∩ Y.
Solution We see that element ‘Geeta’ is the only element common to both. Hence,
X ∩ Y = {Geeta}.
Example 17 Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and B = { 2, 3, 5, 7 }. FindA ∩ B and
hence show that A ∩ B = B.
Solution We have A ∩ B = { 2, 3, 5, 7 } = B. We
note that B ⊂ A and that A ∩ B = B.
Definition 7 The intersection of two sets A and B
is the set of all those elements which belong to both
A and B. Symbolically, we write
A ∩ B = {x : x ∈ A and x ∈ B}
The shaded portion in Fig 1.5 indicates the
intersection of A and B.
Fig 1.4
Fig 1.5
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16. 16 MATHEMATICS
If A and B are two sets such that A ∩ B = φ, then
A and B are called disjoint sets.
For example, letA= { 2, 4, 6, 8 } and
B = { 1, 3, 5, 7 }. Then A and B are disjoint sets,
because there are no elements which are common to
A and B. The disjoint sets can be represented by
means of Venn diagram as shown in the Fig 1.6
In the above diagram, A and B are disjoint sets.
Some Properties of Operation of Intersection
(i) A ∩ B = B ∩ A (Commutative law).
(ii) ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) (Associative law).
(iii) φ ∩ A = φ, U ∩ A = A (Law of φ and U).
(iv) A ∩ A = A (Idempotent law)
(v) A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) (Distributive law ) i. e.,
∩ distributes over ∪
This can be seen easily from the following Venn diagrams [Figs 1.7 (i) to (v)].
A B
U
Fig 1.6
(i) (iii)
(ii) (iv)
(v)
Figs 1.7 (i) to (v)
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17. SETS 17
Fig 1.8
Fig 1.9
1.10.3 Difference of sets The difference of the sets A and B in this order is the set
of elements which belong to Abut not to B. Symbolically, we writeA – B and read as
“ A minus B”.
Example 18 Let A = { 1, 2, 3, 4, 5, 6}, B = { 2, 4, 6, 8 }. Find A – B and B – A.
Solution We have, A – B = { 1, 3, 5 }, since the elements 1, 3, 5 belong to A but
not to B and B – A = { 8 }, since the element 8 belongs to B and not to A.
We note that A – B ≠ B – A.
Example 19 Let V = { a, e, i, o, u } and
B = { a, i, k, u}. Find V – B and B – V
Solution Wehave,V–B={ e,o},sincetheelements
e, o belong to V but not to B and B – V = { k }, since
the element k belongs to B but not to V.
We note that V – B ≠ B – V. Using the set-
builder notation, we can rewrite the definition of
difference as
A – B = { x : x ∈ A and x ∉ B }
The difference of two sets A and B can be
represented by Venn diagram as shown in Fig 1.8.
The shaded portion represents the difference of
the two sets A and B.
Remark The sets A – B, A ∩ B and B – A are
mutually disjoint sets, i.e., the intersection of any of
these two sets is the null set as shown in Fig 1.9.
EXERCISE 1.4
1. Find the union of each of the following pairs of sets :
(i) X = {1, 3, 5} Y = {1, 2, 3}
(ii) A = [ a, e, i, o, u} B = {a, b, c}
(iii) A = {x : x is a natural number and multiple of 3}
B = {x : x is a natural number less than 6}
(iv) A = {x : x is a natural number and 1 < x ≤6 }
B = {x : x is a natural number and 6 < x < 10 }
(v) A = {1, 2, 3}, B = φ
2. Let A = { a, b }, B = {a, b, c}. Is A ⊂ B ? What is A ∪ B ?
3. If A and B are two sets such that A ⊂ B, then what is A ∪ B ?
4. If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find
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18. 18 MATHEMATICS
(i) A ∪ B (ii) A ∪ C (iii) B ∪ C (iv) B ∪ D
(v) A ∪ B ∪ C (vi) A ∪ B ∪ D (vii) B ∪ C ∪ D
5. Find the intersection of each pair of sets of question 1 above.
6. IfA = { 3, 5, 7, 9, 11 }, B = {7, 9, 11, 13}, C = {11, 13, 15}and D = {15, 17}; find
(i) A ∩ B (ii) B ∩ C (iii) A ∩ C ∩ D
(iv) A ∩ C (v) B ∩ D (vi) A ∩ (B ∪ C)
(vii) A ∩ D (viii) A ∩ (B ∪ D) (ix) ( A ∩ B ) ∩ ( B ∪ C )
(x) ( A ∪ D) ∩ ( B ∪ C)
7. If A = {x : x is a natural number }, B = {x : x is an even natural number}
C = {x : x is an odd natural number}andD = {x : x is a prime number }, find
(i) A ∩ B (ii) A ∩ C (iii) A ∩ D
(iv) B ∩ C (v) B ∩ D (vi) C ∩ D
8. Which of the following pairs of sets are disjoint
(i) {1, 2, 3, 4} and {x : x is a natural number and 4 ≤ x ≤ 6 }
(ii) { a, e, i, o, u } and { c, d, e, f }
(iii) {x : x is an even integer } and {x : x is an odd integer}
9. If A = {3, 6, 9, 12, 15, 18, 21}, B = { 4, 8, 12, 16, 20 },
C = { 2, 4, 6, 8, 10, 12, 14, 16 }, D = {5, 10, 15, 20 }; find
(i) A – B (ii) A – C (iii) A – D (iv) B – A
(v) C – A (vi) D – A (vii) B – C (viii) B – D
(ix) C – B (x) D – B (xi) C – D (xii) D – C
10. If X= { a, b, c, d } and Y = { f, b, d, g}, find
(i) X – Y (ii) Y – X (iii) X ∩ Y
11. If R is the set of real numbers and Q is the set of rational numbers, then what is
R – Q?
12. State whether each of the following statement is true or false. Justify your answer.
(i) { 2, 3, 4, 5 } and { 3, 6} are disjoint sets.
(ii) { a, e, i, o, u } and { a, b, c, d }are disjoint sets.
(iii) { 2, 6, 10, 14 } and { 3, 7, 11, 15} are disjoint sets.
(iv) { 2, 6, 10 } and { 3, 7, 11} are disjoint sets.
1.11 Complement of a Set
Let U be the universal set which consists of all prime numbers and A be the subset of
U which consists of all those prime numbers that are not divisors of 42. Thus,
A = {x : x ∈ U and x is not a divisor of 42 }. We see that 2 ∈ U but 2 ∉ A, because
2 is divisor of 42. Similarly, 3 ∈ U but 3 ∉ A, and 7 ∈ U but 7 ∉ A. Now 2, 3 and 7 are
the only elements of U which do not belong toA. The set of these three prime numbers,
i.e., the set {2, 3, 7} is called the Complement ofAwith respect to U, and is denoted by
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19. SETS 19
A′. So we have A′ = {2, 3, 7}. Thus, we see that
A′ = {x : x ∈ U and x ∉ A }. This leads to the following definition.
Definition 8 Let U be the universal set and A a subset of U. Then the complement of
A is the set of all elements of U which are not the elements of A. Symbolically, we
write A′ to denote the complement of A with respect to U. Thus,
A′ = {x : x ∈ U and x ∉ A }. Obviously A′ = U – A
We note that the complement of a set A can be looked upon, alternatively, as the
difference between a universal set U and the set A.
Example 20 Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 3, 5, 7, 9}. Find A′.
Solution We note that 2, 4, 6, 8, 10 are the only elements of U which do not belong to
A. Hence A′ = { 2, 4, 6, 8,10 }.
Example 21 Let U be universal set of all the students of Class XI of a coeducational
school and A be the set of all girls in Class XI. Find A′.
Solution Since A is the set of all girls, A′ is clearly the set of all boys in the class.
Note If A is a subset of the universal set U, then its complement A′ is also a
subset of U.
Again in Example 20 above, we have A′ = { 2, 4, 6, 8, 10 }
Hence (A′)′= {x : x ∈ U and x ∉ A′}
= {1, 3, 5, 7, 9} = A
Itisclearfromthedefinitionofthecomplementthatforanysubsetoftheuniversal
set U, we have ( A′)′ = A
Now, we want to find the results for ( A ∪ B )′ and A′ ∩ B′ in the followng
example.
Example 22 Let U = {1, 2, 3, 4, 5, 6}, A = {2, 3} and B = {3, 4, 5}.
Find A′, B′ , A′ ∩ B′, A ∪ B and hence show that ( A ∪ B )′ = A′∩ B′.
Solution Clearly A′ = {1, 4, 5, 6}, B′ = { 1, 2, 6 }. Hence A′ ∩ B′ = { 1, 6 }
Also A ∪ B = { 2, 3, 4, 5 }, so that (A ∪ B )′ = { 1, 6 }
( A ∪ B )′ = { 1, 6 } = A′ ∩ B′
It can be shown that the above result is true in general. If A and B are any two
subsets of the universal set U, then
( A ∪ B )′ = A′ ∩ B′. Similarly, ( A ∩ B )′ = A′ ∪ B′ . These two results are stated
in words as follows :
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20. 20 MATHEMATICS
The complement of the union of two sets is
the intersection of their complements and the
complement of the intersection of two sets is the
union of their complements. These are called De
Morgan’s laws. These are named after the
mathematician De Morgan.
The complement A′ of a set A can be represented
by a Venn diagram as shown in Fig 1.10.
The shaded portion represents the complement of the set A.
Some Properties of Complement Sets
1. Complement laws: (i) A ∪ A′ = U (ii) A ∩ A′ = φ
2. De Morgan’s law: (i) (A ∪ B)´ = A′ ∩ B′ (ii) (A ∩ B )′ = A′ ∪ B′
3. Law of double complementation : (A′)′ = A
4. Laws of empty set and universal set φ′ = U and U′ = φ.
These laws can be verified by using Venn diagrams.
EXERCISE 1.5
1. Let U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = { 1, 2, 3, 4}, B = { 2, 4, 6, 8 } and
C = { 3, 4, 5, 6 }. Find (i) A′ (ii) B′ (iii) (A ∪ C)′ (iv) (A ∪ B)′ (v) (A′)′
(vi) (B – C)′
2. If U = { a, b, c, d, e, f, g, h}, find the complements of the following sets :
(i) A = {a, b, c} (ii) B = {d, e, f, g}
(iii) C = {a, c, e, g} (iv) D = { f, g, h, a}
3. Taking the set of natural numbers as the universal set, write down the complements
of the following sets:
(i) {x : x is an even natural number} (ii) { x : x is an odd natural number }
(iii) {x : x is a positive multiple of 3} (iv) { x : x is a prime number }
(v) {x : x is a natural number divisible by 3 and 5}
(vi) { x : x is a perfect square } (vii) { x : x is a perfect cube}
(viii) { x : x + 5 = 8 } (ix) { x : 2x + 5 = 9}
(x) { x : x ≥ 7 } (xi) { x : x ∈ N and 2x + 1 > 10 }
4. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = {2, 4, 6, 8} and B = { 2, 3, 5, 7}. Verify that
(i) (A ∪ B)′ = A′ ∩ B′ (ii) (A ∩ B)′ = A′ ∪ B′
5. Draw appropriate Venn diagram for each of the following :
(i) (A ∪ B)′, (ii) A′ ∩ B′, (iii) (A ∩ B)′, (iv) A′ ∪ B′
6. Let U be the set of all triangles in a plane. If A is the set of all triangles with at
least one angle different from 60°, what is A′?
Fig 1.10
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21. SETS 21
7. Fill in the blanks to make each of the following a true statement :
(i) A ∪ A′ = . . . (ii) φ′ ∩ A = . . .
(iii) A ∩ A′ = . . . (iv) U′ ∩ A = . . .
1.12 Practical Problems on Union and
Intersection of Two Sets
In earlier Section, we have learnt union, intersection
and difference of two sets. In this Section, we will
go through some practical problems related to our
daily life.The formulae derived in this Section will
also be used in subsequent Chapter on Probability
(Chapter 16).
Let A and B be finite sets. If A ∩ B = φ, then
(i) n ( A ∪ B ) = n ( A ) + n ( B ) ... (1)
The elements in A∪ B are either inAor in B but not in both as A∩ B = φ. So, (1)
follows immediately.
In general, if A and B are finite sets, then
(ii) n ( A ∪ B ) = n ( A ) + n ( B ) – n ( A ∩ B ) ... (2)
Note that the sets A – B, A ∩ B and B – A are disjoint and their union is A ∪ B
(Fig 1.11). Therefore
n ( A ∪ B) = n ( A – B) + n ( A ∩ B ) + n ( B – A )
= n ( A – B) + n ( A ∩ B ) + n ( B – A ) + n ( A ∩ B ) – n ( A ∩ B)
= n ( A ) + n ( B ) – n ( A ∩ B), which verifies (2)
(iii) If A, B and C are finite sets, then
n ( A ∪ B ∪ C ) = n ( A ) + n ( B ) + n ( C ) – n ( A ∩ B ) – n ( B ∩ C)
– n ( A ∩ C ) + n ( A ∩ B ∩ C ) ... (3)
In fact, we have
n ( A ∪ B ∪ C ) = n (A) + n ( B ∪ C ) – n [ A ∩ ( B ∪ C ) ] [ by (2) ]
= n (A) + n ( B ) + n ( C ) – n ( B ∩ C ) – n [ A ∩ ( B ∪ C ) ] [ by (2) ]
Since A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ), we get
n [ A ∩ ( B ∪ C ) ] = n ( A ∩ B ) + n ( A ∩ C ) – n [ ( A ∩ B ) ∩ (A ∩ C)]
= n ( A ∩ B ) + n ( A ∩ C ) – n (A ∩ B ∩ C)
Therefore
n ( A ∪ B ∪ C ) = n (A) + n ( B ) + n ( C ) – n ( A ∩ B ) – n ( B ∩ C)
– n ( A ∩ C ) + n ( A ∩ B ∩ C )
This proves (3).
Example 23 If X and Y are two sets such that X ∪ Y has 50 elements, X has
28 elements and Y has 32 elements, how many elements does X ∩ Y have ?
Fig 1.11
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22. 22 MATHEMATICS
Solution Given that
n ( X ∪ Y ) = 50, n ( X ) = 28, n ( Y ) = 32,
n (X ∩ Y) = ?
By using the formula
n ( X ∪ Y ) = n ( X ) + n ( Y ) – n ( X ∩ Y ),
we find that
n ( X ∩ Y ) = n ( X ) + n ( Y ) – n ( X ∪ Y )
= 28 + 32 – 50 = 10
Alternatively, suppose n ( X ∩ Y ) = k, then
n ( X – Y ) = 28 – k , n ( Y – X ) = 32 – k (by Venn diagram in Fig 1.12 )
This gives 50 = n ( X ∪ Y ) = n (X – Y) + n (X ∩ Y) + n ( Y – X)
= ( 28 – k ) + k + (32 – k )
Hence k = 10.
Example 24 In a school there are 20 teachers who teach mathematics or physics. Of
these, 12 teach mathematics and 4 teach both physics and mathematics. How many
teach physics ?
Solution Let M denote the set of teachers who teach mathematics and P denote the
set of teachers who teach physics. In the statement of the problem, the word ‘or’ gives
us a clue of union and the word ‘and’ gives us a clue of intersection. We, therefore,
have
n ( M ∪ P ) = 20 , n ( M ) = 12 and n ( M ∩ P ) = 4
We wish to determine n ( P ).
Using the result
n ( M ∪ P ) = n ( M ) + n ( P ) – n ( M ∩ P ),
we obtain
20 = 12 + n ( P ) – 4
Thus n ( P ) = 12
Hence 12 teachers teach physics.
Example 25 In a class of 35 students, 24 like to play cricket and 16 like to play
football. Also, each student likes to play at least one of the two games. How many
students like to play both cricket and football ?
Solution Let X be the set of students who like to play cricket and Y be the set of
students who like to play football. Then X ∪ Y is the set of students who like to play
at least one game, and X ∩ Y is the set of students who like to play both games.
Given n ( X) = 24, n ( Y ) = 16, n ( X ∪ Y ) = 35, n (X ∩ Y) = ?
Using the formula n ( X ∪ Y ) = n ( X ) + n ( Y ) – n ( X ∩ Y ), we get
35 = 24 + 16 – n (X ∩ Y)
Fig 1.12
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23. SETS 23
Thus, n (X ∩ Y) = 5
i.e., 5 students like to play both games.
Example 26 In a survey of 400 students in a school, 100 were listed as taking apple
juice, 150 as taking orange juice and 75 were listed as taking both apple as well as
orange juice. Find how many students were taking neither apple juice nor orange juice.
Solution Let U denote the set of surveyed students and A denote the set of students
taking apple juice and B denote the set of students taking orange juice. Then
n (U) = 400, n (A) = 100, n (B) = 150 and n (A ∩ B) = 75.
Now n (A′ ∩ B′) = n (A ∪ B)′
= n (U) – n (A ∪ B)
= n (U) – n (A) – n (B) + n (A ∩ B)
= 400 – 100 – 150 + 75 = 225
Hence 225 students were taking neither apple juice nor orange juice.
Example 27 There are 200 individuals with a skin disorder, 120 had been exposed to
the chemical C1
, 50 to chemical C2
, and 30 to both the chemicals C1
and C2
. Find the
number of individuals exposed to
(i) Chemical C1
but not chemical C2
(ii) Chemical C2
but not chemical C1
(iii) Chemical C1
or chemical C2
Solution Let U denote the universal set consisting of individuals suffering from the
skin disorder, A denote the set of individuals exposed to the chemical C1
and B denote
the set of individuals exposed to the chemical C2
.
Here n ( U) = 200, n ( A ) = 120, n ( B ) = 50 and n ( A ∩ B ) = 30
(i) From the Venn diagram given in Fig 1.13, we have
A = ( A – B ) ∪ ( A ∩ B ).
n (A) = n( A – B ) + n( A ∩ B ) (Since A – B) and A ∩ B are disjoint.)
or n ( A – B ) = n ( A ) – n ( A ∩ B ) = 120 –30 = 90
Hence, the number of individuals exposed to
chemical C1
but not to chemical C2
is 90.
(ii) From the Fig 1.13, we have
B = ( B – A) ∪ ( A ∩ B).
and so, n (B) = n (B – A) + n ( A ∩ B)
(Since B – A and A ∩B are disjoint.)
or n ( B – A ) = n ( B ) – n ( A ∩ B )
= 50 – 30 = 20 Fig 1.13
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24. 24 MATHEMATICS
Thus, the number of individuals exposed to chemical C2
and not to chemical C1
is 20.
(iii) The number of individuals exposed either to chemical C1
or to chemical C2
, i.e.,
n ( A ∪ B ) = n ( A ) + n ( B ) – n ( A ∩ B )
= 120 + 50 – 30 = 140.
EXERCISE 1.6
1. If X and Y are two sets such that n ( X ) = 17, n ( Y ) = 23 and n ( X ∪ Y ) = 38,
find n ( X ∩ Y ).
2. If X and Y are two sets such that X ∪ Y has 18 elements, X has 8 elements and
Y has 15 elements ; how many elements does X ∩ Y have?
3. In a group of 400 people, 250 can speak Hindi and 200 can speak English. How
many people can speak both Hindi and English?
4. If S and T are two sets such that S has 21 elements, T has 32 elements, and S ∩ T
has 11 elements, how many elements does S ∪ T have?
5. If X and Y are two sets such that X has 40 elements, X ∪ Y has 60 elements and
X ∩ Y has 10 elements, how many elements does Y have?
6. In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least
one of the two drinks. How many people like both coffee and tea?
7. In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many
like tennis only and not cricket? How many like tennis?
8. In a committee, 50 people speak French, 20 speak Spanish and 10 speak both
Spanish and French. How many speak at least one of these two languages?
Miscellaneous Examples
Example 28 Show that the set of letters needed to spell “ CATARACT ” and the
set of letters needed to spell “ TRACT” are equal.
Solution Let X be the set of letters in “CATARACT”. Then
X = { C, A, T, R }
Let Y be the set of letters in “ TRACT”. Then
Y = { T, R, A, C, T } = { T, R, A, C }
Since every element in X is in Y and every element in Y is in X. It follows that X = Y.
Example 29 List all the subsets of the set { –1, 0, 1 }.
Solution Let A = { –1, 0, 1 }. The subset of A having no element is the empty
set φ. The subsets of A having one element are { –1 }, { 0 }, { 1 }. The subsets of
A having two elements are {–1, 0}, {–1, 1} ,{0, 1}. The subset of A having three
elements of AisAitself. So, all the subsets ofA are φ, {–1}, {0}, {1}, {–1, 0}, {–1, 1},
{0, 1} and {–1, 0, 1}.
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25. SETS 25
Example 30 Show that A ∪ B = A ∩ B implies A = B
Solution Let a ∈ A. Then a ∈ A ∪ B. Since A ∪ B = A ∩ B , a ∈ A ∩ B. So a ∈ B.
Therefore, A ⊂ B. Similarly, if b ∈ B, then b ∈ A ∪ B. Since
A ∪ B = A ∩ B, b ∈ A ∩ B. So, b ∈ A. Therefore, B ⊂ A. Thus, A = B
Example 31 For any sets A and B, show that
P ( A ∩ B ) = P ( A ) ∩ P ( B ).
Solution Let X ∈ P ( A ∩ B ). Then X ⊂ A ∩ B. So, X ⊂ A and X ⊂ B. Therefore,
X ∈ P ( A ) and X ∈ P ( B ) which implies X ∈ P ( A ) ∩ P ( B). This gives P ( A ∩ B )
⊂ P ( A ) ∩ P ( B ). Let Y ∈ P ( A ) ∩ P ( B ). Then Y ∈ P ( A) and Y ∈ P ( B ). So,
Y ⊂ Aand Y ⊂ B. Therefore, Y ⊂ A ∩ B, which implies Y ∈ P (A ∩ B ). This gives
P ( A ) ∩ P ( B ) ⊂ P ( A ∩ B)
Hence P ( A ∩ B ) = P ( A ) ∩ P ( B ).
Example 32 A market research group conducted a survey of 1000 consumers and
reported that 720 consumers like productAand 450 consumers like product B, what is
the least number that must have liked both products?
Solution Let U be the set of consumers questioned, S be the set of consumers who
liked the product A and T be the set of consumers who like the product B. Given that
n ( U ) = 1000, n ( S ) = 720, n ( T ) = 450
So n ( S ∪ T ) = n ( S ) + n ( T ) – n ( S ∩ T )
= 720 + 450 – n (S ∩ T) = 1170 – n ( S ∩ T )
Therefore, n ( S ∪ T ) is maximum when n ( S ∩ T ) is least. But S ∪ T ⊂ U implies
n ( S ∪ T ) ≤ n ( U ) = 1000. So, maximum values of n ( S ∪ T ) is 1000. Thus, the least
value of n ( S ∩ T ) is 170. Hence, the least number of consumers who liked both products
is170.
Example 33 Out of 500 car owners investigated, 400 owned car A and 200 owned
car B, 50 owned both A and B cars. Is this data correct?
Solution Let U be the set of car owners investigated, M be the set of persons who
owned car A and S be the set of persons who owned car B.
Given that n ( U ) = 500, n (M ) = 400, n ( S ) = 200 and n ( S ∩ M ) = 50.
Then n ( S ∪ M ) = n ( S ) + n ( M ) – n ( S ∩ M ) = 200 + 400 – 50 = 550
But S ∪ M ⊂ U implies n ( S ∪ M ) ≤ n ( U ).
This is a contradiction. So, the given data is incorrect.
Example 34 A college awarded 38 medals in football, 15 in basketball and 20 in
cricket. If these medals went to a total of 58 men and only three men got medals in all
the three sports, how many received medals in exactly two of the three sports ?
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26. 26 MATHEMATICS
Solution Let F, B and C denote the set of men who
received medals in football, basketball and cricket,
respectively.
Then n ( F ) = 38, n ( B ) = 15, n ( C ) = 20
n (F ∪ B ∪ C ) = 58 and n (F ∩ B ∩ C ) = 3
Therefore, n (F ∪ B ∪ C ) = n ( F ) + n ( B )
+ n ( C ) – n (F ∩ B ) – n (F ∩ C ) – n (B ∩ C ) +
n ( F ∩ B ∩ C ),
gives n ( F ∩ B ) + n ( F ∩ C ) + n ( B ∩ C ) = 18
Consider the Venn diagram as given in Fig 1.14
Here, a denotes the number of men who got medals in football and basketball only, b
denotes the number of men who got medals in football and cricket only, c denotes the
number of men who got medals in basket ball and cricket only and d denotes the
number of men who got medal in all the three. Thus, d = n ( F ∩ B ∩ C ) = 3 and
a + d + b + d + c + d = 18
Therefore a + b + c = 9,
which is the number of people who got medals in exactly two of the three sports.
Miscellaneous Exercise on Chapter 1
1. Decide, among the following sets, which sets are subsets of one and another:
A = { x : x ∈ R and x satisfy x2
– 8x + 12 = 0 },
B = { 2, 4, 6 }, C = { 2, 4, 6, 8, . . . }, D = { 6 }.
2. In each of the following, determine whether the statement is true or false. If it is
true, prove it. If it is false, give an example.
(i) If x ∈ A and A ∈ B , then x ∈ B
(ii) If A ⊂ B and B ∈ C , then A ∈ C
(iii) If A ⊂ B and B ⊂ C , then A ⊂ C
(iv) If A ⊄ B and B ⊄ C , then A ⊄ C
(v) If x ∈ A and A ⊄ B , then x ∈ B
(vi) If A ⊂ B and x ∉ B , then x ∉ A
3. Let A, B, and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. Show
that B = C.
4. Show that the following four conditions are equivalent :
(i) A ⊂ B(ii) A – B = φ (iii) A ∪ B = B (iv) A ∩ B = A
5. Show that if A ⊂ B, then C – B ⊂ C – A.
6. Assume that P ( A ) = P ( B ). Show that A = B
7. Is it true that for any sets A and B, P ( A ) ∪ P ( B ) = P ( A ∪ B )? Justify your
answer.
Fig 1.14
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27. SETS 27
8. Show that for any sets A and B,
A = ( A ∩ B ) ∪ ( A – B ) and A ∪ ( B – A ) = ( A ∪ B )
9. Using properties of sets, show that
(i) A ∪ ( A ∩ B ) = A (ii) A ∩ ( A ∪ B ) = A.
10. Show that A ∩ B = A ∩ C need not imply B = C.
11. Let A and B be sets. If A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X for some set
X, show that A = B.
(Hints A = A ∩ ( A ∪ X ) , B = B ∩ ( B ∪ X ) and use Distributive law )
12. Find sets A, B and C such that A ∩ B, B ∩ C and A ∩ C are non-empty
sets and A ∩ B ∩ C = φ.
13. In a survey of 600 students in a school, 150 students were found to be taking tea
and 225 taking coffee, 100 were taking both tea and coffee. Find how many
students were taking neither tea nor coffee?
14. In a group of students, 100 students know Hindi, 50 know English and 25 know
both. Each of the students knows either Hindi or English. How many students
are there in the group?
15. In a survey of 60 people, it was found that 25 people read newspaper H, 26 read
newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T,
8 read both T and I, 3 read all three newspapers. Find:
(i) the number of people who read at least one of the newspapers.
(ii) the number of people who read exactly one newspaper.
16. In a survey it was found that 21 people liked productA, 26 liked product B and
29 liked product C. If 14 people liked productsAand B, 12 people liked products
C and A, 14 people liked products B and C and 8 liked all the three products.
Find how many liked product C only.
Summary
This chapter deals with some basic definitions and operations involving sets. These
are summarised below:
A set is a well-defined collection of objects.
A set which does not contain any element is called empty set.
A set which consists of a definite number of elements is called finite set,
otherwise, the set is called infinite set.
Two sets A and B are said to be equal if they have exactly the same elements.
A set A is said to be subset of a set B, if every element of A is also an element
of B. Intervals are subsets of R.
A power set of a set A is collection of all subsets of A. It is denoted by P(A).
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28. 28 MATHEMATICS
The union of two sets Aand B is the set of all those elements which are either
in A or in B.
The intersection of two sets A and B is the set of all elements which are
common. The difference of two setsAand B in this order is the set of elements
which belong to A but not to B.
The complement of a subsetA of universal set U is the set of all elements of U
which are not the elements of A.
For any two sets A and B, (A ∪ B)′ = A′ ∩ B′ and ( A ∩ B )′ = A′ ∪ B′
If A and B are finite sets such that A ∩ B = φ, then
n (A ∪ B) = n (A) + n (B).
If A ∩ B ≠ φ, then
n (A ∪ B) = n (A) + n (B) – n (A ∩ B)
Historical Note
The modern theory of sets is considered to have been originated largely by the
German mathematician Georg Cantor (1845-1918). His papers on set theory
appeared sometimes during 1874 to 1897. His study of set theory came when he
was studying trigonometric series of the form a1
sin x + a2
sin 2x + a3
sin 3x + ...
He published in a paper in 1874 that the set of real numbers could not be put into
one-to-one correspondence wih the integers. From 1879 onwards, he publishd
several papers showing various properties of abstract sets.
Cantor’s work was well received by another famous mathematician Richard
Dedekind (1831-1916). But Kronecker (1810-1893) castigated him for regarding
infinite set the same way as finite sets. Another German mathematician Gottlob
Frege, at the turn of the century, presented the set theory as principles of logic.
Till then the entire set theory was based on the assumption of the existence of the
set of all sets. It was the famous Englih Philosopher Bertand Russell (1872-
1970 ) who showed in 1902 that the assumption of existence of a set of all sets
leads to a contradiction. This led to the famous Russell’s Paradox. Paul R.Halmos
writes about it in his book ‘Naïve Set Theory’ that “nothing contains everything”.
The Russell’s Paradox was not the only one which arose in set theory.
Many paradoxes were produced later by several mathematicians and logicians.
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29. SETS 29
As a consequence of all these paradoxes, the first axiomatisation of set theory
was published in 1908 by Ernst Zermelo.Another one was proposed byAbraham
Fraenkel in 1922. John Von Neumann in 1925 introduced explicitly the axiom of
regularity. Later in 1937 Paul Bernays gave a set of more satisfactory
axiomatisation. A modification of these axioms was done by Kurt Gödel in his
monograph in 1940. This was known as Von Neumann-Bernays (VNB) or Gödel-
Bernays (GB) set theory.
Despite all these difficulties, Cantor’s set theory is used in present day
mathematics. In fact, these days most of the concepts and results in mathematics
are expressed in the set theoretic language.
— —
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