How many relations are there on the power set of $X = \{\emptyset\}$?

Question

Considering the fact that a set has 2^n elements in the power set hence it must have 2^1=2 elements and hence 2^(2*2) [2^(n*n)] which is equal to 2^4 but the answer says it must be 2^16.

Answer 1

Question 1: What is the definition of a relation and how many are there for a given finite set? You will see that the number of relations only depends on the size of the set, not on the actual elements.

Question 2: How many elements does the power set of $X$ have?

Answer 2

A relation on a set $A$ is a subset of $A\times A$. Since your set $X$ has a single element, $\mathcal{P}(X)$ has two elements and $\mathcal{P}(X)\times\mathcal{P}(X)$ has $4$ elements. Therefore $\mathcal{P}(X)\times\mathcal{P}(X)$ has $2^4=16$ and only $16$ subsets. So, the answer is $16$.