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If set X consists of three elements then the number of elements in the power set of power set of X is
(a) ${{3}^{3}}$
(b) ${{2}^{3}}$
(c) ${{3}^{8}}$
(d) ${{2}^{8}}$

Last updated date: 13th Jul 2024
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Hint: In this question, we have to find the number of elements in the power set of the power set of X. It is assumed that in set X, there are a total of three elements. All we have to do is find its power set. And then again we have to find the power set of that power set. After that, calculate the total number of elements in the resultant power set.

Complete step by step answer:
Now, let’s discuss the question.
As we all know that set is a collection of objects or elements, grouped in the curly braces. For example: Set A = {a, b, c, d, e} In this ‘A’ is the name of the set and ‘a’, ‘b’, ‘c’, ‘d’ and ‘e’ are the elements of set A. If we say that we have set A of even numbers such as A = {2, 4, 6, 8, 10, 12, 14} and we have a set B = {2, 4, 6}, so we can say that B is a subset of A i.e. $B\subseteq A$ which means A contains all the elements of B and A will be called a superset of B. If we are given an empty set such as A = { }, any set cannot be empty because by default it contains an element which is null. So this set will also be called a null set. In subsets of a particular set, we have to write all the possible elements and pairs which are possible in the given set. For example we are given a set D = {a, b}, so the subsets of D = {$\phi $ , a, b, {a, b}, {b, a}}. The number of subsets will be calculated as ${{2}^{n}}$ where n is the number of elements in the given set. So this is also called the power set.
For this question, let’s assume that set X = {1, 2, 3}
So the power set of this set will be ${{2}^{n}}$and place n = 3, we will get ${{2}^{3}}$. As per question, we have to find the power of this power set also. It will be ${{\left( 2 \right)}^{{{\left( 2 \right)}^{3}}}}$. So,
$\Rightarrow {{\left( 2 \right)}^{{{\left( 2 \right)}^{3}}}}\Leftrightarrow {{2}^{8}}$

So, the correct answer is “Option d”.

Note: If we are given a power to a number that means that number has to be multiplied as many times has the power. For example: we will express ${{3}^{4}}$ as $3\times 3\times 3\times 3$ which is equal to 81. If the power is zero on any number, the result will be 1. The null value is not counted while calculating power. It is only calculated for the number of elements present in the set except null.