Let $X$be a set containing $n$ elements. The number of reflexive relations that can be defined on $X$ is 

Let $X=\left\{x_1,x_2,\dots x_{\mathrm{n}}\right\}$ The set $X\times X$ contains $n^2$ elements. 

A reflexive relation $R$ on $X$ must contain $\left(x_{\mathrm{i}},x_{\mathrm{i}}\right)$ for all $1\le i\le n\text{,}$

i.e.,R must contain the n elements $\left(x_1,x_1\right),\left(x_2,x_2\right)$ $\dots \left(x_0,x_0\right)$ and any subset of the set containing remaining  $n^2–n$ elements. Therefore, the number of reflexive relations that can be defined on $X$ is $2^{n^2-n}$