Set Relations Quick Question

Question

Can someone please explain how this answer was reached? I know that relation of A is just A * A but wouldn't that just be $4^{2}? $

Let A = {1, 2, 3, 4}. How many relations are there on a set A?

Solution: $2^{4^2} = 2^{16} = 131,072$

Answer 1

By definition, a relation on $A$ is a subset of $A^2$. Since $|A^2|=16$, the number of subsets of $A^2$ is $2^{16}$, which means there are $2^{16}$ relations on $A$.