Ouestion B$1$

$($a$)$ Using De Morgan's theorem and Boolean algebra simplify the following expressions to

the simplest logic expressions

$($i$)(\frac{?}{\text{b}\text{a}\text{r}}(A)+B)\frac{A+B+D}{b}ar(D)$

$4\%$

$($ii$)$ ABC$+\frac{A}{b}ar(\frac{?}{\text{b}\text{a}\text{r}}\frac{B}{b}ar(\frac{?}{\text{b}\text{a}\text{r}}(A)))$

$($b$)$ You are given the Karnaugh map presented in Figure B$1\mathrm{.}1.$ Find the minimum sum of

products expression that corresponds to the Karnaugh map and then implement it using

the minimum number of $2-$inputs NOR gates.

Figure B$1\mathrm{.}1$

$($c$)$ Implement the logic expression in Question B$1($b$)$ using the dual $4$x$1$ multiplexer

depicted in Figure B$1\mathrm{.}2.$ The multiplexer has two enable inputs $($i$.$e$.,$ G$0,$ G$1)$ for each

section and two select inputs $($i$.$e$.,$ S$1,$ S$2).$ G$0$ and G$1$ enable inputs control the upper

and the lower $41$ multiplexers, respectively. Do not use input $D$ as an input to the

selectors of the multiplexer.

$($d$)$ Is it possible to implement the logic expression in Question B$1($b$)$ using a single $4$x$1$

multiplexer? If yes, show the design. Justify the answer.