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Basics of Set Theory-Part 1(methods of representing a set )

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 Π 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 : x2 – 2 = 0 and x is rational number}. 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

Example 1   Write the solution set of the equation   x2 – 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 x2 < 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 = n2, 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 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. 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, 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}

 

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