How to prove/What laws to use for $(A\times B)∩(B\times A)$ $=$ $(A∩B)\times (A∩B)$. I'm stuck on where to start for my homework. I'm trying to re-write the left side using one of the sets laws. But I'm either blind or I just have no idea.
Is there any easier to start or is using the laws the best way to go about it?

First, welcome to MSE. I hope you enjoy your stay.
With regards to your problem I would recommend using the definition of equality of sets. Specifically two sets $C$ and $D$ are said to be equal if $C\subseteq D$ and $D\subseteq C$. That is, if $x\in C$ then $x\in D$ and if $x\in D$ then $x\in C$.
You might try this for your problem. Let $p$ be a point in $(A\times B)\cap(B\times A)$. Then $p\in A\times B$ and $p\in B\times A$. We can rewrite this statement by first denoting $p$ by $(x,y)$. For $(x,y)$ to be in $A\times B$ we must have that $x\in A$ and $y\in B$. However, we know that $(x,y)$ is also in $B\times A$ so $x\in B$ and $y\in A$. Combining these two statements we have that $x\in A\cap B$ and $y\in A\cap B$. Therefore $p=(x,y)\in(A\cap B)\times(A\cap B)$, which by definition means that $(A\times B)\cap(B\times A)\subseteq (A\cap B)\times(A\cap B)$.
You prove the reverse inclusion in a similar way. Good luck, and welcome again.
