How to prove this equality using boolean algebra?

Question

I have approximately no idea on how to solve the following problem, so any help would very much be appreciated:

$$x' y'+ x y = (x y' + x' y)'$$

I can't figure out how to prove the equalities using boolean algebra
Thank you for your help!

Answer 1

Here's one way. First, use DeMorgan's law: $$ (xy' + x'y)' = (xy')'(x'y)' = (x' + y)(x + y') $$ From there, you can expand the product using the distributive property, noting that $x'x$, which means "$x$ and not $x$", is $0$ (i.e. false). $$ (x' + y)(x + y') = (x'x) + xy + x'y' + (yy') = xy + x'y' $$