Sets — Exercise 5(E)

Write the cardinal number of each of the following sets:

(i) A = { 0, 1, 2, 4 }

(ii) B = { -3, -1, 1, 3, 5, 7 }

(iii) C = { }

(iv) D = { 3, 2, 2, 1, 3, 1, 2 }

(v) E = { Natural numbers between 15 and 20 }

(vi) F = { Whole numbers from 8 to 14 }

Answer

(i) A = { 0, 1, 2, 4 } has 4 elements.

Hence, n(A) = 4.

(ii) B = { -3, -1, 1, 3, 5, 7 } has 6 elements.

Hence, n(B) = 6.

(iii) C = { } is the empty set, which has no element.

Hence, n(C) = 0.

(iv) D = { 3, 2, 2, 1, 3, 1, 2 } = { 1, 2, 3 } has 3 elements.

Hence, n(D) = 3.

(v) The natural numbers between 15 and 20 are 16, 17, 18 and 19. So, E = { 16, 17, 18, 19 }, which has 4 elements.

Hence, n(E) = 4.

(vi) The whole numbers from 8 to 14 are 8, 9, 10, 11, 12, 13 and 14. So, F = { 8, 9, 10, 11, 12, 13, 14 }, which has 7 elements.

Hence, n(F) = 7.

Given:

A = { Natural numbers less than 10 }

B = { Letters of the word 'INVENTION' }

C = { Squares of the first four whole numbers }

D = { Odd numbers divisible by 2 }

Find:

(i) n(A)

(ii) n(B)

(iii) n(C)

(iv) n(D)

Answer

(i) The natural numbers less than 10 are 1, 2, 3, 4, 5, 6, 7, 8 and 9. So, A = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }, which has 9 elements.

Hence, n(A) = 9.

(ii) The letters of the word 'INVENTION' are I, N, V, E, N, T, I, O, N. Writing each letter only once, we get I, N, V, E, T, O. So, B = { I, N, V, E, T, O }, which has 6 elements.

Hence, n(B) = 6.

(iii) The first four whole numbers are 0, 1, 2 and 3. Their squares are 0² = 0, 1² = 1, 2² = 4 and 3² = 9. So, C = { 0, 1, 4, 9 }, which has 4 elements.

Hence, n(C) = 4.

(iv) No odd number is divisible by 2, so D = { }, which has no element.

Hence, n(D) = 0.

State true or false for each of the following. Correct the wrong statement.

(i) If A = { 0 }, then n(A) = 0.

(ii) n(∅) = 1.

(iii) If T = { a, l, a, h, b, d, h }; then n(T) = 5

(iv) If B = { 1, 5, 51, 15, 5, 1 }, then n(B) = 6.

Answer

(i) A = { 0 } has one element, namely 0. So, n(A) = 1, not 0.

False; the correct statement is n(A) = 1.

(ii) The empty set has no element. So, n(∅) = 0, not 1.

False; the correct statement is n(∅) = 0.

(iii) T = { a, l, a, h, b, d, h } = { a, l, h, b, d }, which has 5 elements. So, n(T) = 5.

True

(iv) B = { 1, 5, 51, 15, 5, 1 } = { 1, 5, 51, 15 }, which has 4 elements. So, n(B) = 4, not 6.

False; the correct statement is n(B) = 4.