Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8} which of the following may be considered as universal set(s) for all the three sets A, B and C
(i) {0,1, 2, 3, 4, 5, 6} (ii) Φ (iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} (iv) {1, 2, 3, 4, 5, 6, 7, 8}

Solution:

An universal set should contain all the sets as its subsets with respect to the context of the problem.

For example, in our problem, each of the sets A, B, and C should be a subset of the universal set.

(i) We see that

A ⊂ {0, 1, 2, 3, 4, 5, 6}

B ⊂ {0, 1, 2, 3, 4, 5, 6}

C ⊄ {0, 1, 2, 3, 4, 5, 6}

Therefore, the set {0, 1, 2, 3, 4, 5, 6} cannot be the universal set for the sets A, B, and C.

(ii) A ⊄ Φ, B ⊄ Φ C ⊄ Φ

Therefore, Φ cannot be the universal set for sets A, B, and C.

(iii) We see that

A ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

B ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

C ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Therefore, the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is a universal set for the sets A, B, and C.

(iv) We see that

A ⊂ {1, 2, 3, 4, 5, 6, 7, 8}

B ⊂ {1, 2, 3, 4, 5, 6, 7, 8}

C ⊄ {1, 2, 3, 4, 5, 6, 7, 8}

Therefore, the set {1, 2, 3, 4, 5, 6, 7, 8} cannot be the universal set for the sets A, B, and C

NCERT Solutions Class 11 Maths Chapter 1 Exercise 1.3 Question 9

Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universal set (s) for all the three sets A, B and C (i) {0, 1, 2, 3, 4, 5, 6} (ii) φ (iii) {0,1,2,3,4,5,6,7,8,9,10} (iv)  {1,2,3,4,5,6,7,8}

Summary:

Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, we are asked to find which of the given sets may be considered as universal set (s) for all the three sets A, B and C. We found that
(i) {0, 1, 2, 3, 4, 5, 6} is NOT a universal set
(ii) φ is NOT a universal set
(iii) {0,1,2,3,4,5,6,7,8,9,10} is a universal set
iv) {1,2,3,4,5,6,7,8} is NOT a universal set