Question

For a set A, the power set of A is denoted by 2^A. If A = {5, {6}, {7}}, which of the following options are TRUE?

I. Φ ϵ 2^A

II. Φ ⊆ 2^A

III. {5, {6}} ∈ 2^A

IV. {5, {6}} ⊆ 2^A
1. I and III only
2. II and III only
3. I, II and III only
4. I, II and IV only

Answer 1

Correct Answer - Option 3 : I, II and III only

Concepts:

In mathematics, the power set (or powerset) of any set S is the set of all subsets of S, including the empty set and S itself.

Explanation:

A = {5, {6}, {7}}

Power set of A = 2^A = {Φ, {5}, {{6}}, {{7}}, {5, {6}}, {5, {7}}, {{6}, {7}}, {5, {6}, {7}}}

Statement I

Φ is element of power set of A. Therefore, Φ ϵ 2^A.

Statement II.

Power set of A consists of all subsets of A and from the definition of a subset, ϕ is a subset of any set.

Therefore, Φ ⊆ 2^A

Statement III

{5, {6}} is element of power set of A. Therefore, {5, {6}} ϵ 2^A.

Statement IV

{5, {6}} is element of power set of A. Therefore, {{5, {6}}} ⊆ 2^A.

Hence statement IV is false.

Therefore option 3 is correct.