Construct a truth table for each of the following Boolean expressions.

F$=$x$*$y$+$z $$x $+$y $$

G$=$a$*$b$*$c $+$a $*$c$*$d $+$c$*$d$+$a$*$b $*$c $*$d

Simplify the following Boolean statements. You do not need to specify which identities you are using. Note that $($x$+$z$)$ is really $(($x$+$z$))$

A $*$C$*$D$+$A $*($A$*$B$)+$A$*$C$*$D

$($x$*$y$+$z$)+($x$+$y$*$z$)+($x $+$y$)$ a$.$

b$.$

F $=$ x$+$z $,$ G $=$ x $*$y$+$z show $($and reduce$)$ each of F $,$ G $,$ F$*$G$,($F$*$G$),$ and F $+$G $.$ When done, you should have $($F$*$G$)=$F $+$G $$ to prove that DeMorgan$$s Law holds for F and G$.$ Note: you will need to apply DeMorgan$$s Law in doing your reductions. NOTE: for $($F$*$G$),$ you may reach a point where you have to take one of your terms and add $*1$ to continue to simplify it to the result of F $+$G $.$

Prove the following two expressions are equal by showing that

the left$-$hand side and right$-$hand side simplify to the same expression

both sides have the same truth table results

w$*$y$+$w$*$x$*$z $+($x$+$y $)+($w$+$y $)+($x $+$z$)=($y $*($x $+$z$))$

Note that the right hand side does not have a w$,$ for the truth table this means that the results are the same for w $=0$ and w $=1,$ that is$,$ the RHS expression should still have $16$ rows where it doesn$$t matter what w is$.$

Use sum of products to obtain the Boolean expression that matches the truth table below and then simplify the expression as much as possible.

X Y Z Function

$0001$

$0011$

$0100$

$0111$

$1001$

$1011$

$1100$

$1111$

Draw the circuits for the expressions $($unsimplified$)$ from question number $1.$