**Basics of Set Theory**

**What is a Set**?

*A set can be defined as collection of objects in a well defined manner. All such well defined objects are called elements of set.*

*Sets are usually denoted by capital alphabets e.g. A,B,C,D,E,F,G…… and elements as a,b,c,d,e,f……or numerals as 1,2,3,4,5,6…..or in any other suitable defined manner.*

If ‘a’ is elemant of set ‘A’, it is written as *a * Î A. If ‘*b*’ is not an element of a set A, we write *b *Ï A and read these as “a belongs to A” and “*b *does not belong toA”.

Symbol Î is read as “*belongs to*” or “*element of *”.

Elements of a set are usually written inside braces and separated by coma e.g. a set S of vowels of English alphabets is S={a,e,i,o,u}. In this set aÎS whereas b Ï S.

**There are two methods of representing a set :**

(i) Roster or tabular form

(ii) Set-builder form.

**Roster form**

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 factors of 44 is {1, 2, 4, 11, 22, 44}.

(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 show that the list of odd numbers continue indefinitely.

In a set, the order in which the lements are listed is immaterial e.g.

S={1, 2, 4, 11, 22, 44} or

S={4, 1, 44, 22,11, 2}

is one and the same.

Usually elements are not repeated in a set e.g. a set of letters of word ‘CAPITAL’ is {C, A, P, I, T, L}. Here ‘A’ has not been repeated in the list of elements.

**Set builder form**

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 B, we write

B = {*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 B 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 *< 9} is read as “the set of all *x *such that *x *is a natural number and *x *lies between 3 and 9. Hence, the numbers 4, 5, 6, 7 and 8 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 44}

B= {*y *: *y *is a vowel in the English alphabet}

C= {*z *: *z *is an odd natural number}

**Empty set or Null set **

Sometimes there may not be any element in a set, such sets are called ‘Empty Set’ or ‘Null Set’ and denoted by the symbol f 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 *: *x*^{2} – 2 = 0 and *x *is rational number}. B is the empty set because the equation *x*^{2} – 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 *: *x*^{2 }= 4, *x *is odd }. Then D is the empty set, because the equation *x*^{2} = 4 is not satisfied by any odd value

**Example 1 **Write the solution set of the equation x^{2 }– 3x * +*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 *x*^{2} < 30} in the roster form

**Solution **The required numbers are 1, 2, 3, 4, 5 . So, the given set in the roster form

is {1, 2, 3, 4, 5 }.

**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 *= *n*^{2}, where *n *Î **N**}. N is set of all natural numbers.

**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 begins from 1 and do not exceed 6. Hence, in the set-builder form, the given set is

{ x : x=(n/n+1)_{__}where n is a natural number and 1 ≤ n ≥ 6 }

**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 *x*^{2} – 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, *x*^{2} – 9 = 0 implies *x *= 3, –3 and so (iv) matches (b).

**EXERCISE **

**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.

(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 Î or Ï 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, *x*^{2} ≤ 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}