Why does the Boolean equation A.B' + B = A + B hold?

Question

I have pretty good amount of knowledge in Boolean algebra. However, I struggled with the equality $(1)$ more than I should have.

$$x'z' + z = x' + z\tag{1}$$

How is it that this holds, algebraically? I can assure you that I've tried it enough. I just cannot get it right now.

Thanks.

Answer 1

For any Boolean value $y$, $\color{blue}{1+y = 1}$.

So, $\color{blue}{x'z'+z} = x'z'+z\cdot1 = x'z'+z(1+x') = x'(z+z')+z = \color{blue}{x'+z}$