elementary set theory (cartesian product and symmetric difference proof)

I've figured out the following statement is true but I was wondering how you actually go about proving something like this?

$A \times (B \triangle C) = (A \times B) \triangle (A \times C)$

Solution 1:

A nice approach is by comparing the characteristic functions. Note that :

$(x,y)\in A\times B\Leftrightarrow \left(x\in A\,\mathrm{and}\,y\in B\right)$

so that $1_{A\times B}=1_A\times 1_B$

Also $1_{A\Delta B}=1_A\oplus A_B$ where $\oplus$ denotes mod 2 addition.

Now the required proof is straightforward.