Showing $\lvert\mathscr{B}_X\rvert=2^{n^2}$.

Question

This is Exercise 2.9.11 of Howie's "Fundamentals of Semigroup Theory".

Let $\mathscr{B}_X$ be the set of all binary relations on a set $X$, where $\lvert X\rvert=n$. Show that $\lvert\mathscr{B}_X\rvert=2^{n^2}$.

My Attempt:

For each $(x, y)\in X\times X$, $(x, y)$ is either in a given binary relation on $X$ or it is not. Thus $$\begin{align} \lvert\mathscr{B}_X\rvert&=\lvert\{\text{is in, is not in}\}\rvert^{\lvert X\times X\rvert} \\ &=2^{\lvert X\times X\rvert} \\ &=2^{\lvert X\rvert^2} \\ &=2^{n^2}. \end{align}$$ $\square$

Thoughts:

Whilst, I suppose, my proof is satisfactory, I think it lacks rigour. Would you accept my proof?

Answer 1

Hint. A binary relation on $X$ can be represented by a Boolean $n \times n$ matrix. What is the number of Boolean $n \times n$matrices?