Proof using boolean algebra properties

Question

I need to prove that

$$ x + y = y ~~~\iff~~~ xy = x $$

also need to prove that:

$$ x + y = x ~~~->~~~ x + not(y) = 1 $$

Using algebra boolean properties.

But I have no idea how to solve this problem.

Any ideas? Thanks

Answer 1

Here's a useful principle for the first problem:

Absorption

$x + xy = x$

$x(x+y) = x$

Applied to your problem 1, going from left to right:

Assume $x + y = y$. Then $xy = x(x+y) = x$

Can you go from right to left?

For the second problem, use:

Complement

$x + x' = 1$

$xx'=0$

and

Annihilation

$x + 1 = 1$

$x0=0$