Boolean Algebra proving algebraically simple

Question

$$(X'+Y )(X+Y')=XY+X'Y'$$

I am just wondering how these are equal, and what laws are used to get there

Answer 1

$$ (X'+Y)(X+Y') = XX'+X'Y'+XY+YY' \\ \begin{array}{cl} XX' = 0 & \text{by definition of negation} \\ YY' = 0 & \text{same} \end{array} \\ $$ so $$ (X'+Y)(X+Y') = XY+X'Y' $$ In classical notation it looks like $$ (\neg x \vee y)\wedge(x \vee \neg y) = (\neg x \wedge x )\vee (\neg x \wedge \neg y) \vee (y \wedge x) \vee (y \wedge \neg y) = 0\vee (\neg x \wedge \neg y) \vee (x \wedge y)\vee 0 = \\ = (\neg x \wedge \neg y) \vee (x \wedge y) $$