# How do you compute the number of reflexive relation?

Given a set with $n$ elements

I know that there is $2^{n^2}$ relations, because there are $n$ rows and $n$ columns and it is either $1$ or $0$ in each case, but I don't know how to compute the number of reflexive relation. I am very dumb. Can someone help me go through the thought process?

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A relation is reflexive if and only if every entry on the main diagonal of its matrix is $1$; that’s the only restriction. Fill in $1$’s on the diagonal, and you can put either $0$ or $1$ freely into every other entry in the matrix and have the matrix of a reflexive relation. Thus, the number of reflexive relations on a set of $n$ elements is $2^m$, where $m$ is the number of entries that are not on the diagonal. There are $n^2$ entries altogether, and $n$ of them are on the diagonal, so how many are not on the diagonal? And then how many reflexive relations are there?
Strange way you have to count the number of relations...A relation on a set $\,A\,$ is just a subset of the cartesian product $\,A\times A\,$, and if $\,|A|=n\Longrightarrow |A\times A|=n^2\Longrightarrow\,$ the number of subsets of $\,A\times A\,$ , i.e. $\,|P(A\times A)|\,$ , is $\,2^{|A\times A|}=2^{n^2}\,$...
Now, you need to count all the subsets of $\,A\times A\,$ that contain the diagonal $\,\Delta_A:=\{(a,a)\in A\times A\}\,$ , so...can you take it form here?