Find the sum-of-products expansions the Boolean function F(x, Y, 2) that equals 1 if and only if a)x = 0. b) xy = 0. c) x +y = 0. d)xyz = 0.

a)

$\overline{x}(yz+\overline{y}z+y\overline{z}+\overline{y}\overline{z})=\overline{x}yz+\overline{x}\overline{y}z+\overline{x}y\overline{z}+\overline{x}\overline{y}\overline{z}$

b)

$\overline{xy }= \overline {x}+\overline{y}$

then:

$\overline{xy}z+\overline{xy}\overline{z}=(\overline {x}+\overline{y})z+(\overline{x}+\overline{y})\overline{z}=\overline{x}z+\overline{y}z+\overline{x}\overline{z}+\overline{y}\overline{z}$

c)

$\overline{x +y}=\overline{x}\cdot \overline{y}$

then:

$\overline{x}\cdot \overline{y}\cdot z+\overline{x}\cdot \overline{y}\cdot \overline{z}$

d)

$\overline{xyz}=\overline{xy}+\overline{z}=\overline{x}+\overline{y}+\overline{z}$