Simplifying a boolean expression using the simple laws

Question

I am very confused with the laws of boolean algebra and how to use them.

How am I supposed to simplify this:

$xy + (x+y)z'+y$

to this:

$xz'+y$

Answer 1

Basically, you need to use distribution laws and the fact that $1+a\equiv 1$ for a boolean variable $a$. $$xy+(x+y)z' + y = xy + xz'+yz'+y = xz'+(x+z'+1)y = xz'+1y=xz'+y$$

Answer 2

$$xy + (x +y)z'+ y = \text{ (Distribution)}$$

$$ xy + xz' + yz' + y = \text{ (Absorption)}$$

$$ xz' +y$$

So the really useful rule here is :

Absorption

$p + pq = p$

In your case, the $y$ term absorbs the terms $xy$ and $yz'$