Proof about sets, Cartesian Product

Question

Can someone please explain this answer to me with a graphical representation or just a have a better explanation than this? I would really appreciate it, please, Thanks

Answer 1

I'm VERY late to the party, but here is my quick MSPaint drawing of the situation. I hope it helps!

Everything else is just a detailed write-up of the image above.

Answer 2

Sure. Elements of $A\times(B\cap C)$ look like this: $(a,d)$, where $a\in A$, $d\in B$, and $d\in C$. Elements of $(A\times B)\cap(A\times C)$ look like this: $(a',d')$, where $a'\in A$, $(a',d')\in A\times B$, and $(a',d')\in A\times C$.

On the one hand, if $d\in B$ and $d\in C$ then certainly $(a,d)\in A\times B$ and $(a,d)\in B\times C$. On the other hand, if $(a',d')\in A\times B$ and $(a',d')\in A\times C$ then certainly $d'\in B$ and $d'\in C$. Therefore the sets are equal.

Is there a specific part of the proof that you don't understand? Theorems like this are a bit of a nuisance to prove, but if you work with them enough you'll be able to quickly see why they are true.