No set identity for Cartesian product?

[ainster31]

Homework Statement

Homework Equations

I have to use these set identities:

The Attempt at a Solution

Pretty sure this is impossible because there's no identity for the Cartesian product.

 

[kduna]

Just go at it the old fashion way.

Suppose (a, d) $\in$ A X (B $\cup$ C). Then a $\in$ A. Also d $\in$ B or d $\in$ C. So (a,d) $\in$ (A X B) or (a,d) $\in$ (A X C).

Thus (a,d) $\in$ (A X B) $\cup$ (A X C).

Therefore A X (B $\cup$ C) $\subseteq$ (A X B) $\cup$ (A X C).

Proving the subset goes the other way follows similarly.