Correct way to simplify $x+xy$ and $xy+xz+x'z+zx$

Question

For Boolean variables, which of the following is correct?

$$x+xy= x(1+y)= xy,$$

or

$$x+xy= (x+y)(x+x)= (x+y)x?$$

And as a second problem,

$$xy+xz+x'z+zx = xy+z(x+x')+zx= xy+z(1+x)= xy+zx.$$

Does it end here or can it be simplified further?

Answer 1

Assuming that you are dealing with Boolean variables, you are making the following mistake:

• For any Boolean variable $b$ you have $1+b \color{blue}{=} 1$ (hence $1+b \color{red}{\neq b}$).

Hence, your expressions simplify differently: $$x+xy = x(1+y)= x1 = x$$ and $$xy+xz+x'z+zx = xy+z(x+x')+zx = xy+z\underbrace{(1+x)}_{=1} = xy+z$$