Laws of Logic Expression

Question

Going through yet another practice exam and I'm stumped on the below question. Any help is appreciated.

What laws were used to reduce this:

$E = x\cdot (x+y)\cdot \overline{y}$

to this:

$E = x\cdot\overline{y}$

Answer 1

It is always true that $x \cdot x = x$ (idempotence), $x \cdot \bar{x} = 0$ (complementation), $x\cdot 0=0$ (anihilation), and $x+0=x$ (identity).

Then by distribution and the above : $x(x+y) \bar{y} ~{= x \cdot x \cdot \bar{y} + x \cdot y \cdot \bar{y} \\ = x \cdot \bar{y}}$.

Answer 2

The fact that $x\cdot(x+y)=x$ is called Absorption, which is a basic law of logic

In the event that Absorption was not provided to you, here is Absorption derived from other basic rules:

$$x\cdot(x+y) \overset{Identity}= (x+0)\cdot (x+y)\overset{Distribution}=x+0\cdot y\overset{Annihilation}= x+0 \overset{Identity}= x$$