Question

If A is a void set then n[P(P(P(A)))] is
(a) 0
(b) 1
(c) 3
(d) 4

Answer 1

Hint: We will use \[{{2}^{n}}\] to calculate the number of elements n(P(A)) and hence successively step by step will reach to calculate n[P(P(P(A)))]. If A is the set then the set of all the subsets of A is the power set of A. The power set of A is represented by P(A).

Complete step-by-step answer:
Before proceeding with the question, we should understand the concept of sets and power sets. The set of all the subsets is the power set of that particular set, including null set and itself also. If A is the set then the set of all the subsets of A is the power set of A. The power set of A is represented by P(A). Example- Z = {1, 3, 5}. Here, the number of the elements of Z is 3. The subsets of Z are { }, {1}, {3}, {5}, {1, 3}, {3, 5}, {1, 5}, {1, 3, 5}. The power set of Z is the set of all the subsets of Z. We will list all the subsets then enclose them in the curly braces “{ }”. P(Z) = { { }, {1}, {3}, {5}, {1, 3}, {3, 5}, {1, 5}, {1, 3, 5} }.
Cardinality is the number of elements of a set. The number of elements of a power set is the number of subsets. The number of elements is represented by Z. As we know that the formula of calculating the number of subsets is \[{{2}^{n}}\].
We can write it as: P(Z)\[={{2}^{n}}..........(1)\]
Here in the question it is mentioned that the number of elements in A is 0 that is a void set. So putting n equal to zero in equation (1) we get,
Hence, n(P(A))\[={{2}^{0}}=1.......(2)\]
From equation (2) we see that P(A) has one element.
Now, n(P(P(A))\[={{2}^{1}}=2........(3)\]
From equation (3) we see that P(P(A)) has two elements.
Hence n[P(P(P(A)))] \[={{2}^{2}}=4\].
Hence option (d) is the correct answer.

Note: Here knowing the formula of the number of elements in a power set is important. In such types of questions students may forget to include the empty set as a subset and that will give us the wrong answer.