p. 1

e-528-529 sector-7 dwarka new delhi-110075 nr ramphal chowk and sector 9 metro station ph 011-47350606 m 7838010301-04 www.eduproz.in educate anytime anywhere greetings for the day about eduproz we at eduproz started our voyage with a dream of making higher education available for everyone since its inception eduproz has been working as a stepping-stone for the students coming from varied backgrounds the best part is ­ the classroom for distance learning or correspondence courses for both management mba and bba and information technology mca and bca streams are free of cost experienced faculty-members a state-of-the-art infrastructure and a congenial environment for learning are the few things that we offer to our students our panel of industrial experts coming from various industrial domains lead students not only to secure good marks in examination but also to get an edge over others in their professional lives our study materials are sufficient to keep students abreast of the present nuances of the industry in addition we give importance to regular tests and sessions to evaluate our students progress students can attend regular classes of distance learning mba bba mca and bca courses at eduproz without paying anything extra our centrally air-conditioned classrooms well-maintained library and well-equipped laboratory facilities provide a comfortable environment for learning honing specific skills is inevitable to get success in an interview keeping this in mind eduproz has a career counselling and career development cell where we help student to prepare for interviews our dedicated placement cell has been helping students to land in their dream jobs on completion of the course eduproz is strategically located in dwarka west delhi walking distance from dwarka sector 9 metro station and 4minutes drive from the national highway students can easily come to our centre from anywhere delhi and neighbouring gurgaon haryana and avail of a quality-oriented education facility at apparently no extra cost why choose edu proz for distance learning?

[close]

p. 2

· · · · · · · · · edu proz provides class room facilities free of cost in eduproz class room teaching is conducted through experienced faculty class rooms are spacious fully air-conditioned ensuring comfortable ambience course free is not wearily expensive placement assistance and student counseling facilities edu proz unlike several other distance learning courses strives to help and motivate pupils to get high grades thus ensuring that they are well placed in life students are groomed and prepared to face interview boards mock tests unit tests and examinations are held to evaluate progress special care is taken in the personality development department have a good day karnataka state open university ksou was established on 1st june 1996 with the assent of h.e governor of karnataka as a full fledged university in the academic year 1996 vide government notification no/edi/uov/dated 12th february 1996 karnataka state open university act ­ 1992 the act was promulgated with the object to incorporate an open university at the state level for the introduction and promotion of open university and distance education systems in the education pattern of the state and the country for the co-ordination and determination of standard of such systems keeping in view the educational needs of our country in general and state in particular the policies and programmes have been geared to cater to the needy karnataka state open university is a ugc recognised university of distance education council dec new delhi regular member of the association of indian universities aiu delhi permanent member of association of commonwealth universities acu london uk asian association of open universities aaou beijing china and also has association with commonwealth of learning col karnataka state open university is situated at the north­western end of the manasagangotri campus mysore the campus which is about 5 kms from the city centre has a serene atmosphere ideally suited for academic pursuits the university houses at present the administrative office academic block lecture halls a well-equipped library guest house

[close]

p. 3

cottages a moderate canteen girls hostel and a few cottages providing limited accommodation to students coming to mysore for attending the contact programmes or termend examinations bt0063-unit-01-set theory unit 1 set theory structure 1.1 introduction objectives 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 sets and their representations the empty set finite and infinite sets equal and equivalent sets subsets power set universal set venn diagrams complement of a set operations on sets 1.12 applications of sets 1.13 1.14 cartesian product of sets summary 1.15 terminal questions 1.16 answers

[close]

p. 4

1.1 introduction the concept of set is basic in all branches of mathematics it has proved to be of particular importance in the foundations of relations and functions sequences geometry probability theory etc the study of sets has many applications in logic philosophy etc the theory of sets was developed by german mathematician georg cantor 1845 ­ 1918 a.d he first encountered sets while working on problems on trigonometric series in this unit we discuss some basic definitions and operations involving sets objectives at the end of the unit you would be able to · · · understand the concepts of sets perform the different operations on sets write the power set of a given set 1.2 sets and their representations in every day life we often speak of collection of objects of a particular kind such as pack of cards a herd of cattle a crowd of people cricket team etc in mathematics also we come across various collections for example collection of natural numbers points in plane prime numbers more specially we examine the collections 1 2 3 4 5 odd natural numbers less than 10 i.e 1 3 5 7 9 the rivers of india the vowels in the english alphabet namely a e i o u prime factors of 210 namely 2 3 5 and 7 the solutions of a equation x2 ­ 5x 6 0 viz 2 and 3 we note that each of the above collections is a well defined collection of objects in the sense that we can definitely decide whether a given object belongs to a given collection or not for example we can say that the river nile does not belong to collection of rivers of india on the other hand the river ganga does belong to this collection however the following collections are not well defined 1 2 3 4 the collection of bright students in class xi of a school the collection of renowned mathematicians of the world the collection of beautiful girls of the world the collection of fat people for example in ii above 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:

[close]

p. 5

1 objects elements and members of a set are synonymous terms these are undefined 2 sets are usually denoted by capital letters a b c x y z etc 3 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 is used to denote the phrase `belongs to thus we write if b is not an element of a set a we write but and read `b does not belong to a thus in the set v of vowels in the english alphabet in the set p of prime factors of but there are two methods of representing a set i ii roster or tabular form set builder form i in roster form all the elements of a set are listed the elements 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 note that in roster form the order in which the elements are listed is immaterial thus the above set can also be represented as {l 3 7 21 2 6 14 42 b the set of all vowels in the english alphabets is {a e i o u c the set of odd natural numbers is represented by {1 3 5 the three dots tell us that the list is endless 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 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 each of them is a vowel in the english alphabet and no other letter possesses this property denoting this set by v we write v {x x is a vowel in the english alphabet it may be observed that we describe the set by using a symbol x for elements of the set any other symbol like the letters y z etc could also be used in place of x after the sign of `colon write

[close]

p. 6

the characteristic property possessed by the elements of the set and then enclose the 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 following description of a 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 3 x 10 hence the numbers 4 5 6 7 8 and 9 are the elements of set a if we denote the sets described above in a b and c in roster form by a b and c respectively then a b and 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 write the set of all vowels in the english alphabet which precede q solution the vowels which precede q are a e i o thus a {a e i o is the set of all vowels in the english alphabet which precede q example write the set of all positive integers whose cube is odd solution the cube of an even integer is also an even integer so the members of the required set can not be even also cube of an odd integer is odd so the members of the required set are all positive odd integers hence in the set builder form we write this set as {x x is an odd positive integer or equivalently as {2k 1 k 0 k is an integer example write the set of all real numbers which can not be written as the quotient of two integers in the set builder form solution we observe that the required numbers can not be rational numbers because a rational number is a number in the form where p q are integers and q 0 thus these must be real and irrational hence in set builder form we write this set as {x x is real and irrational}

[close]

p. 7

example write the set in the set builder form solution each member in the given set has the denominator one more than the numerator also the numerators begin from 1 and do not exceed 6 hence in the set builder form the given set is example match each of the sets on the left described in the roster form with the same set on the right described in the set builder form i ii iii iv l i t e {0 b a {x x is a positive integer and is a divisor of 18 {x x is an integer and x2 ­ 9 0 c {x x is an integer and x 1 1 {1 2 3 6 9 18 {3 ­ 3 d {x x is a letter of the word little solution since in d there are six letters in the word little and two letters t and l 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 so iii matches a finally x2 ­ 9 0 implies x 3 ­3 so iv matches b example write the set {x x is a positive integer and x2 40 in the roster form 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 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 consider the set {x x is an integer x2 1 0 we know that there is no integer whose square is ­1 so the above set has no elements we now define set b as follows:

[close]

p. 8

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 hence the set b contains no element at all definition 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 the empty set is denoted by the symbol we give below a few examples of empty sets i let p {x 1 x 2 x is a natural number then p is an empty set because there is no natural number between 1 and 2 ii let q {x x2 ­ 2 0 and x is rational then q is the empty set because the equation x2 2 0 is not satisfied by any rational number x iii let r {x x is an even prime number greater than 2 then r is the empty set because 2 is the only even prime number iv let s {x x2 4 and x is an odd integer then s is the empty set because equation x2 4 is not satisfied by any value of x which is an odd integer 1.4 finite and infinite sets let a {1 2 3 4 5 b {a b c d e f and c {men in the world we observe that a contains 5 elements and b contains 6 elements how many elements does c contain as it is we do not know the exact 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 a we mean the number of distinct elements of the set and we denote it by na if na is a natural number then a is a finite set otherwise the set a is said to be an infinite set for example consider the set n of natural numbers we see that nn i.e the number of elements of n is not finite since there is no natural number which equals nn we thus say that the set of natural number is an infinite set definition a set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite we shall denote several set of numbers by the following symbols:

[close]

p. 9

n zqrz q r the set of natural numbers the set of integers the set of rational numbers the set of real numbers the set of positive integers the set of positive rational numbers the set of positive real numbers we consider some examples 1 2 3 4 let m be the set of days of the week then m is finite q the set of all rational numbers is infinite let s be the set of solution s of the equation x2 ­ 16 0 then s is finite let g be the set of all 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 always possible to write all the elements of an infinite set within braces because the number of elements of such a set is not finite however we represent some of the infinite sets in the roster form by writing a few elements which clearly indicate the structure of the set followed or preceded by three dots for instance {1 2 3 4 is the set of natural numbers {1 3 5 7 9 is the set of odd natural numbers and ­ 3 ­2 ­1 0 1 2 3 is the set of integers but the set of real numbers cannot be described in this form because the elements of this set do not follow any particular pattern 1.5 equal and equivalent 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 the sets a and b are said to be equal clearly the two sets have exactly the same elements definition 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 1 let a {1 2 3 4 and b {3 1 4 2 2 then a b.

[close]

p. 10

3 let a be the set of prime numbers less than 6 and p the set of prime factors of 30 obviously the set a and p are equal since 2 3 and 5 are the only prime factors of 30 and are less than 6 let us consider two sets l {1 2 3 4 and m {1 2 3 8 each of them has four elements but they are not equal definition two finite sets a and b are said to be equivalent if they have the same number of elements we write a b for example let a {a b c d e and b {1 3 5 7 9 then a and b are equivalent sets obviously all equal sets are equivalent but all equivalent sets are not equal example find the pairs of equal sets if any giving reasons a {0 b {x x 15 and x 5 c {x x ­ 5 0 d {x:x2 25 e {x x is a positive integral 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 therefore a a c a d a e b but none of the other sets are empty hence b c b d and b b e c {5 but hence c d since e {5 c e d {­5 5 and e {5 therefore d e thus the only pair of equal sets is c and e 1.6 subsets consider the sets s and t where s denotes the set of all students in your school and t denotes the set of all students in your class we note that every element of t is also an element of s we say that t is a subset of s definition if every element of a set a is also an element of a set b then a is called a subset of b or a is contained in b we write it as a b if at least one element of a does not belong to b then a is not a subset of b we write it as 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 in a then we shall also have b have a b and b a which implies a b a in this case a and b are the same sets so that we

[close]

p. 11

it follows from the definition that every set a is a subset of itself i.e a has no elements we agree to say that examples a since the empty set is a subset of every set we now consider some 1 the set q of rational numbers is a subset of the set r of real numbers and we write q r 2 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 b and b 3 let a {1 3 5 and b {x x is an odd natural number less than 6 then a a and hence a b 4 let a {a e i o u b {a b c d then a is not a subset of b also b is not a subset of a we write a b and b a 5 let us write down all the subsets of the set {1 2 we know is a subset of every set so is a subset of {1 2 we see that {1 {2 and {l 2 are also subsets of {1,2 thus the set {1,2 has in all four subsets viz {1 {2 and {1,2 definition 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 a superset of a for example a {1 2 3 is a proper subset of b {1 2 3 4 definition if a set a has only one element we call it a singleton thus {a is a singleton 1.7 power set in example v of section 1.6 we found all the subsets of the set {1 2 viz {1 {2 and {1 2 the set of all these four subsets is called the power set of {1 2 definition the collection of all subsets of a set a is called the power set of a it is denoted by pa in pa every element is a set example v of section 1.6 if a {1 2 then pa {1 {2 {1,2 also note that n[pa 4 22 in general if a is a set with na m then it can be shown that n[pa 2m m na 1.8 universal set

[close]

p. 12

if in any particular context of sets we find a set u which contains all the sets under consideration as subsets of u then set u is called the universal set we note that the universal set is not unique for example for the set z of all integers the universal set can be the set q of rational numbers or for that matter the set r of real numbers for another example in the context of human population studies the universal set consists of all the people in the world example consider the following sets a {1 3 b {1 5 9 c {1 3 5 7 9 insert the correct symbol i b ii a b iii a c iv b c solution 1 b as is a subset of every set or between each pair of sets 2 a b as 3 a and 3 b 3 a 4 b c as 1 3 a also belongs to c c as each element of b also belongs to c example let a {1 2 3 4 b {1 2 3 and c {2 4 find all sets x such that i x b and x c ii x a and x b solution i x b means that x is a subset of b and the subsets of b are {1 {2 {3 {1,2 {1,3 c means that x is a subset of c and the subsets of c are {2 {4 b and x c means that x is a subset of both b and c {2,3 and {1,2,3 x and {2 4 thus we note that x hence x {2 ii x a x b means that x is a sub set of a but x is not a subset of b so x is one of these {4 {1,2,4 {2,3,4 {l,3,4 {1,4 {2,4 {3,4 {1,2,3,4 note a set can easily have some elements which are themselves sets for example {1 {2,3 4 is a set having {2,3 as one element which is a set and also elements 1,4 which are not sets.

[close]

p. 13

example 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 c 1 2 here a b as a {1 and b c implies b c but a c as 1 a and 1 c note that an element of a set can never be a subset of it 1.9 diagrams venn most of the relationships between sets can be represented by means of diagrams figures representing sets in the form of enclosed region in the plane are called venn diagrams named after british logician john venn 1834 1883 a.d the universal set u is represented by the interior of a rectangle.other sets are represented by the interior of circles fig 1.1 fig 1.1 is a venn diagram representing sets a and b such that a b fig 1.2

[close]

p. 14

in fig.1.2 u {1 2 3 10 is the universal set of which a {2,4,6,8,10 and b {4,6 are subsets it is seen that b a the reader will see an extensive use of the venn diagrams when we discuss the operations on sets 1.10 complement of a set let the universal set u be the set of all prime numbers let a be the subset of u which consists of all those prime numbers that are not divisors of 42 thus a {g x x u and x is not a divisor of 42 we see that 2 u but 2 a because 2 is a divisor of 42 similarly 3 u but 7 a now 2 3 and 7 are the only elements of u which do not belong to a the set of these three prime numbers i.e the set {2 3 7 is called the complement of a with respect to u and is denoted by 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 u but 3 a and 7 definition let u be the universal set and a is a subset of u then the complement of a with respect to w.r,t u 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 it can be represented by venn diagram as fig 1.3 the shaded portion in fig 1.3 represents a example 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

[close]

p. 15

example let u be the universal set of all the students of class xi of a co-educational school let a be the set of all girls in the class xl find a solution as a is the set of all girls hence a is the set of all boys in the class 1.11 operations on sets in earlier classes you learnt how to perform the operations of addition subtraction multiplication and division on numbers you also studied certain properties of these operations namely commutativity associativity distributivity etc we shall now define operations on sets and examine their properties henceforth we shall refer all our sets as subsets of some universal set a 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 of a as well as the elements of b the common elements being taken only once the symbol is used to denote the union thus we can define the union of two sets as follows definition 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 symbolically we write `a union b {x:x a or x b and usually read as the union of two sets can be represented by a venn diagram as shown in fig 1.4 fig 1.4 the shaded portion in fig 1.4 represents a b b example let a {2 4 6 8 and b {6 8 10 12 find a

[close]