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Basics of Set Theory – Part 2( Finite,Infinite & Equal Sets)

Finite and Infinite Sets

Let A = {1, 2, 3, 4, 5}, B = {a, b, c, d, e, g}

A set having definit number of elements is called a finite set and a set having indefinite number of elements is called infinite set. Empty set is a finite set.

Set A = {M, A, H, E, S, H} is finite set with 5 elements and Set B = { 2, 3, 5, 7, 11, 13} is finite set with 6 elements.  Number of elements of a set means the number of distinct elements of the set and is denoted as n (S). If n (S)

is a natural number, then S is non-empty finite set.

Consider a  Set C of all odd numbers. We see that the number of elements of this

set is not finite since there are infinite number of odd numbers. We say that the set

of odd  numbers is an infinite set. The sets A and B given above are finite sets

and n(A) = 5, n(B) = 6 .

Example 6   State which of the following sets are finite or infinite :

(i) {x : x Î N and (x – 1) (x –2) = 0}                     N is set of natural numbers.

(ii) {x : x Î N and x2 = 4}

(iii) {x : x Î N and 2x –1 = 0}

(iv) {x : x Î N and x is prime}

(v) {x : x Î 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 = f.  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.

Equal Sets

If each and every element of one set is same as that of the other set then both sets are equal.

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. Then A and P are equal, since 2, 3 and 5 are the only prime factors of

30 and also these are less than 6.

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

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 Î 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 = f but none of the other sets are empty. Therefore B ≠ C, B ≠ D

and B ≠ E. Also C = {5} but –5 Î 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 Î Z and n2 ≤ 4}  where Z is set of all integers

and B = {x : x Î 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 Î A and 0 Ï B,  A and B are not equal sets.

EXERCISE   

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

(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}

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