Question

Power set of empty set has exactly __________ subset.

1. 1

2. 2

3. 0

4. 3

Answer 1

Hint: Power set of any set(S) includes the empty set as its subset and set S itself also. Number of subset in power set of set S having n element is ${{2}^{n}}$

Complete step-by-step solution -

In general, a power set is a set of all subsets of a given set.

For example if we have set S as S=\left\\{ 1,2 \right\\}

As set S has 2 elements , so number of subset in power set of S is ${{2}^{2}}=4$

So we can write subset of set S as given below:

\left\\{ {} \right\\},\left\\{ 1 \right\\},\left\\{ 2 \right\\},\left\\{ 1,2 \right\\}

For the given question,

So we can see the power set includes the empty set and the given set itself as a subset.

So in the given question we have an empty set {ф}.

So its power set should have one subset which is an empty set itself.

P(ф)= {ф}.

n = 1.

Number of subset in power set of set S having n element is ${{2}^{n}}$

As set has 1 element which is null , so number of subset in power set is ${{2}^{1}}=2$

So option B is correct.

Note: As we discussed the number of subset in the power set given by ${{2}^{n}}$ where n is the number of elements in a given set. The empty number contains a null element which is n=1. So the number of subset in the power set of an empty set is ${{2}^{1}}=2$. We can justify the answer by this way also.