K$-$ Map

$42.$ Use a Karnaugh map to reduce each expression to a minimum SOP form:

$($a$)A+\frac{B}{b}ar(C)+CD$

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

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

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

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