# 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}

A universal set is a set which contains all the elements of its subsets. So if there are two sets with some members then the universal set containing those two sets, will have elements will all the members of both sets without repetition.

(i) Given: A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}

Now, let D = {0, 1, 2, 3, 4, 5, 6}

Since, 8 belongs to C then its universal set must contain 8, but D does not contain 8.

∴ D is not a universal set for A, B, C.

(ii) Given: A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}

Now, let D = ϕ

Since, D is an empty set it does not contain any element

∴ D is not a universal set for A, B, C.

(iii) Given: A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}

Now, let D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Since, all the members of A, B, C belongs to D.

∴D is a universal set for A, B, C.

(iv) Given: A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}

Now, let D = {1, 2, 3, 4, 5, 6, 7, 8}

Since, 0 belongs to C but it is not a member of D.

∴ D is not a universal set.

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