boolean logic simplify

Question

To prove:

$(X+Y)(X'+Z) = XZ + X'Y$

I try to simply $(X+Y)(X'+Z)$ to $XZ + X'Y + YZ$

then I have no idea how to simply further.

Thanks in advance!!

Answer 1

Write $YZ = (X+X^\prime)YZ = XYZ+X^\prime YZ$, then $$\begin{array}{rcl} XZ+X^\prime Y+YZ &=& XZ+X^\prime Y +XYZ+X^\prime YZ \\ &=& XZ(1+Y)+X^\prime Y(1+Z) \\ &=& XZ+X^\prime Y. \end{array}$$

Answer 2

$$\begin{align} (X + Y)(X' + Z) & = (X+Y)X' + (X+Y)Z \\ \\ & = \underbrace{X'X}_{= 0} + X'Y + XZ + YZ \\ \\ & = X'Y + XZ + YZ \\ \\ & = X'Y + XZ + \underbrace{(X+ X')}_{= 1}YZ \\ \\ & = X'Y + XZ + XYZ + X'YZ \\ \\ & = X'Y\underbrace{(1 + Z)}_{= 1} + XZ\underbrace{(1 + Y)}_{= 1} \\ \\ &= X'Y + XZ \end{align}$$