Apply DeMorgan's theorems to:

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

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

Prove that the complement of $A^{\text{'}}B+A{B}^{\text{'}}$ is $A^{\text{'}}{B}^{\text{'}}+AB.$

Simplify the following expressions by using Boolean algebra.

$($a$)A(A^{\text{'}}+AB)$

$($b$)A{B}^{\text{'}}C+A^{\text{'}}BC+A^{\text{'}}{B}^{\text{'}}C$

$($c$)A^{\text{'}}{B}^{\text{'}}C\frac{+}{b}ar((A+B+C^{\text{'}}))+A^{\text{'}}{B}^{\text{'}}{C}^{\text{'}}D$