Proof Cartesian products and sets

Question

Suppose A and B are subsets of a universal set E. Prove that (E × E) \ (A × B) = ((E \ A) × E) ∪ (E × (E \ B)).

Is (E × E) \ (A × B) = ((E\A) ×(E\B)), any tips would be appreciated thanks.

Answer 1

No, $(E\times E)\setminus(A\times B)\neq(E\setminus A)\times (E\setminus B)$. An easy way to see this is to take $E=\mathbb R$, $A=B=[0,1]$ and draw a picture.

To help with the actual statement, suppose $(x,y)\in(E\times E)\setminus(A\times B)$. Then $(x,y)\in E\times E$ and $(x,y)\notin A\times B$. Thus $x\in E\setminus A$ or $y\in E\setminus B$.

Can you take it form here?