Write an expression that is a $1$ when one or more of its variables $,$ and $E)$ are $0$ s $.$

Write an expression that is a $0$ when one or more of its variables $,$ and $C)$ are $0$ s $.$

Evaluate the following operations:

$($a$)0+0+0+0$

$($b$)0+0+0+1$

$($c$)1+1+1+1$

$($d$)1*1+0*0+1$

$($e$)1*0*1*0$

$($f$)1*0+1*0+0*1+0*1$

Find the values of the variables that make each product term $1$ and each sum term $0.$

$($a$)$ ABC

$($b$)A+B+C$

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

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

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

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

Find the value of $x$ for all possible values of the variables.

$($a$)x=A+B+C$

$($b$)x=(A+B)C$

$($c$)x=(A+B)(\frac{?}{\text{b}\text{a}\text{r}}(B+C))$

$($d$)x=(A+B)+(\frac{?}{\text{b}\text{a}\text{r}}(AB+BC))$

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

Section $4-2$ Laws and Rules of Boolean Algebra

$7.$ Identify the law of Boolean algebra upon which each of the following equalities is based:

$($a$)A+AB+$ABC$\frac{+}{b}ar(ABCD)\frac{?}{b}=ar(ABCD)+$ABC$+AB+A$

$($b$)A\frac{+}{b}ar(AB)+$ABC$\frac{+}{b}ar(ABCD)\frac{?}{b}=ar(DCBA)+$CBA$\frac{+}{b}ar(BA)+A$

$($c$)AB(CD\frac{+}{b}ar(CD)+EF\frac{+}{b}ar(EF))=$ABCD$+A\frac{B}{b}ar(CD)+$ABEF$+A\frac{B}{b}ar(EF)$

$8.$ Identify the Boolean rule$($s$)$ on which each of the following equalities is based:

$($a$)\frac{?}{\text{b}\text{a}\text{r}}(\frac{?}{\text{b}\text{a}\text{r}}(AB+CD))\frac{+}{b}ar(EF)=AB+CD\frac{+}{b}ar(EF)$

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

$($c$)A(BC+BC)+AC=A(BC)+AC$

$($d$)AB(C\frac{+}{b}ar(C))+AC=AB+AC$

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

$($f$)$ ABC$\frac{+}{b}ar(AB)\frac{+}{b}ar(ABC)D=$ABC$\frac{+}{b}ar(AB)+D$

Section $4-3$ DeMorgan's Theorems

$9.$ Apply DeMorgan's theorems to each expression:

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

$($b$)\frac{?}{\text{b}\text{a}\text{r}}((\frac{?}{\text{b}\text{a}\text{r}}(A))B)$

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

$($d$)\frac{?}{\text{b}\text{a}\text{r}}(ABC)$

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

$($f$)\frac{?}{\text{b}\text{a}\text{r}}(AB)\frac{+}{b}ar(CD)$

$($g$)\frac{?}{\text{b}\text{a}\text{r}}(AB+CD)$

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