Question

What is the cardinality of the power set of an empty set?
A) $0$
B) $1$
C) $2$
D) Infinity

Answer 1

First we must know, what is cardinality? So, cardinality refers to the number of elements there are in the set. For example, if there is a set such as, S = \left\\{ {1,2,4,8,16} \right\\}. So, there are $5$ elements in the set $S$. So, the cardinality of the set is $5$. So, to solve this question, we have to first find how many members are there in the power set of an empty set. To find the number of power sets, we will use the formula, ${2^n}$, where $n$ is the number of elements in the original set. Then we can easily conclude the cardinality of the set.

Complete step-by-step solution:
We are to find the cardinality of the power set of an empty set.
We know, an empty set has no elements in it.
So, the number of elements in an empty set is $0$.
Therefore, the formula we will use to find the number of power sets is ${2^n}$.
Therefore, the number of power sets of empty sets is, ${P_n} = {2^0} = 1$.
Therefore, the number of elements in the power set of the empty set is $1$.
Power set of empty sets P\left( \phi \right) = \left\\{ \phi \right\\}.
Therefore, the cardinality of the power set of empty set is $1$, the correct option is B.

Note: In this question, we may make a mistake that we may think the power set of an empty set is also empty. We don’t take the null value as a set. We used to denote null as a value for an empty set, but in the case of a power set we have to consider an empty set as a set, that is a null set as a set.