Question

If the cardinal number of the set A is 1, then the cardinal number of the power set $P\left( A \right)$ is:
A. 0
B. 1
C. 2
D. 3

Answer 1

Hint: This question can be answered by first explaining what cardinal number of a set means. We then use the formula for the cardinal number for a power set as ${{2}^{n}}.$ Here, n stands for the cardinal number of the set which in this case is 1. Substituting, we get the solution.

Complete step by step solution:
The cardinal number of a set is defined as the number of distinctive elements in the set. In simple terms, this is nothing but the number of distinct terms in the set which also represents the size of the set. Here, in this question we have the cardinal number of the set as 1 which represents the number of distinctive elements in the set. This is also basically the size of the set.
Cardinal number of a power set $P$ is given by the formula ${{2}^{n}}.$ Here, the value of n is nothing but the cardinal number of the set.
We are required to find the cardinal number of the power set $P\left( A \right)$ , given the cardinal number of the set A as 1. This can be done just by substituting the value of n as 1 in the above equation. Therefore, we find the cardinal number of the power set $P\left( A \right)$ ,
$\Rightarrow {{2}^{n}}$
Substituting the value of n as 1,
$\Rightarrow {{2}^{1}}$
Any number raised to the power 1 is nothing but the number itself.
\[\Rightarrow 2\]
Hence, the cardinal number of the power set $P\left( A \right)$ is 2.

Note: Students need to know the concept of cardinal numbers for sets in order to solve this question easily. If we have the set itself given instead of the cardinal number, we need to be careful while counting the number of elements and choose only the distinctive elements.