$2\mathrm{.}2$ Simplify the following Boolean expressions to a minimum number of literals:

$($a$){?}^{4}xy+x{y}^{\text{'}}+x^{\text{'}}y$

$($c$)xy(x^{\text{'}}+y^{\text{'}}{)}^{\text{'}}$

$($b$)(x+y)(x^{\text{'}}+y)$

$2\mathrm{.}3$ Simplify the following Boolean expressions to a minimum number of literals:

$(a{)}^{**}xyz+x^{\text{'}}y+xy{z}^{\text{'}}$

$(c{)}^{**}(x+y{)}^{\text{'}}(x^{\text{'}}+y^{\text{'}})$

$($b$)*x^{\text{'}}yz+xz$

$(e{)}^{**}(y{z}^{\text{'}}+x^{\text{'}}w)(x{y}^{\text{'}}+z{w}^{\text{'}})$

$($d$){?}^{**}xy+x(wz+wz)$

$(9)(x^{\text{'}}+z^{\text{'}})(x+y^{\text{'}}+z^{\text{'}})$

$2\mathrm{.}4$ Reduce the following Boolean expressions to the indicated number of literals:

$($a$)x^{\text{'}}{y}^{\text{'}}z+x^{\text{'}}yz+x{y}^{\text{'}}$

to three literals

$($b$)(w^{\text{'}}+y)(w^{\text{'}}+y^{\text{'}})(w+x+yz)$

to four literals

$2\mathrm{.}5$ Draw logic diagrams of the circuits that implement the original and simplified expressions

in Problem $2\mathrm{.}2($c$),($e$),$ and $($f$).$

$2\mathrm{.}6$ Draw logic diagrams of the circuits that implement the original and simplified expressions

in Problem $2\mathrm{.}3($a$),($c$),$ and $($f$).$

$2\mathrm{.}7$ Draw logic diagrams of circuits that imptement the original and simplified expressions in

Problem $2\mathrm{.}4(a),(c),$ and $($e$).$

$2\mathrm{.}8$ Find the complement of $F=x^{\text{'}}y+y{z}^{\text{'}}$; then show that $F{F}^{\text{'}}=0$ and $F+F^{\text{'}}=1.$

$2\mathrm{.}9.)$ Find the complement of the following expressions:

$($a$)*x{y}^{\text{'}}z+x^{\text{'}}yz+xyz$

$($b$)x^{\text{'}}{y}^{\text{'}}{z}^{\text{'}}+x^{\text{'}}y{z}^{\text{'}}+x{y}^{\text{'}}{z}^{\text{'}}+x{y}^{\text{'}}z+xy{z}^{\text{'}}$

$($c$)(x{y}^{\text{'}}+w^{\text{'}}z)(w{x}^{\text{'}}+y{z}^{\text{'}})$

$2\mathrm{.}10$ Given the Boolean funclions $F_1$ and $F_2,$ show that

$($a$)$ The Boolepn function $E=F_1+F_2$ contains the sum of the minterns of $F_1$ and $F_2.$

$($b$)$ The Booleth function $G=F_1{F}_2$ contains only the misterms that are common to $F_1$

and $F_2.$

$2\mathrm{.}11$ List the truth table of the function: and draw corresponding circuits.

$($a$)F=xz+x{y}^{\text{'}}+y{z}^{\text{'}}$

$($b$)F=a^{\text{'}}{b}^{\text{'}}{c}^{\text{'}}+a{b}^{\text{'}}c+a^{\text{'}}b{c}^{\text{'}}+ab{c}^{\text{'}}$