How many reflexive relations on $P(A)$

Question

There exist a set $A = \{1,2,3\}$. How many reflexive relations are there on $P(A)$? I don't even know how to begin (beside writting down the members of $P(A)$). Thank you

Answer 1

Let $X$ be any finite set. Then a relation is a subset of $X \times X$ and it is reflexive if it contains $\Delta_X = \{(x,x) \:|\: x \in X\}$. Thus we are looking for the number of subsets of $X \times X$ containing $\Delta_X$ and this is the same as looking for the number of subsets of $(X \times X) - \Delta_X$. This set has $|X|^2 - |X|$ elements and so it has $2^{|X|^2 - |X|}$ subsets. Now just plug in $X = P(A)$.