Please help solving some Boolean Algebra problems

Question

I want to know how these problems are solved. Thank you very much.

i) x·y + y·z + x'·z = x.y + x'·z ii) (x + y)·(x' + z) = x·z + x'·y

Answer 1

For the first one,

$$\begin{align*} xy+yz+x'z&=xy+(x+x')yz+x'z\\ &=xy+xyz+x'yz+x'z\\ &=xy(1+z)+x'z(1+y)\;; \end{align*}$$

can you finish it from there?

For the second,

$$\begin{align*} (x+y)(x'+z)&=(x+y)x'+(x+y)z\\ &=xx'+yx'+xz+yz\\ &=x'y+xz+yz\;; \end{align*}$$

now use the first one after renaming some of the variables.

Answer 2

Well for i) you don't need yz cos xy+x'z <=yz and for ii) when you multiply it you will get similar to i)