Using Boolean algebra, simplify the following expressions: $($a$)($A $+$ B$)($A $+$ C$)($b$)$ AB $+$ ABC $+$ ABCD $+$ ABCDE $($c$)$ AB $+$ ABC $+$ A $($d$)($A $+$ A$)($AB $+$ ABC$)($e$)$ AB $+($A $+$ B$)$C $+$ AB

Using Boolean algebra, simplify the following expressions:

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

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

$($c$)AB\frac{+}{b}ar(AB)C+A$

$($d$)(A\frac{+}{b}ar(A))(AB+A\frac{B}{b}ar(C))$

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