For all problems: Show all work. State any assumptions made.

Simplify the following expressions to minimum$-$cost Boolean expressions.

$($a$)(s\frac{+}{b}ar(t)+u)(r+s+t+u)(\frac{?}{\text{b}\text{a}\text{r}}(r)+s+t+u)$

$($b$)\frac{?}{\text{b}\text{a}\text{r}}(v)w+\frac{v}{b}ar(w)+vw$

Using DeMorgan's Theorem, convert the Boolean expression $\frac{?}{\text{b}\text{a}\text{r}}((a+b)c+()(b+c))$ to

an equivalent expression in which the only NOT operations are on single variables.

Use Venn diagrams to pr$$ve whether or not

$x\frac{+}{b}ar(y)z=(x\frac{+}{b}ar(y))(x+z)$