Answer to Question #153335 in Discrete Mathematics for sai

Question

Find the sum of product expansion of the Boolean function f(x,y,z) = (x+z)y

Answer 1

$F(x,y,z)=(x+z)y$

To determine the sum of product expansion of $F$, we could use boolean identities or construct a table that determines all values of $F$. I will be using boolean identities.

$F(x,y,z)=(x+z)y$

$=xy+zy \text{ (Distributive Law) }\\ =xy.1+zy.1 \text{ (Identity Law) }\\ =xy(z+ \bar{z})+zy(x+\bar{x}) \text{ (Unit Property) }\\ =xyz+xy\bar{z}+zyx+zy\bar{x} \text{(Distributive Law) }\\ =xyz+xy\bar{z}+xyz+\bar{x}yz \text{ (Commutative Law) }\\ =xyz+xy\bar{z}+\bar{x}yz \text{(Idempotent Law) }$