Complete the following exercises for combinational logic design and show your work.

Complete a truth table that has $3$ inputs $($A$,$ B$,$ and C$)$ and one output $($F$).$ F is asserted whenever $A$ is asserted. $F$ is also asserted whenever $C$ is deasserted. $F$ is deasserted in all other cases.

Determine the sum$-$of$-$products equation for $F.$ Show the terms instead of the minterm abbreviations $($show ABC instead of $m7).$

Determine the product$-$of$-$sums equation for $F.$ Show the terms instead of the maxterm abbreviations $($show $A^{\text{'}}+B^{\text{'}}+C^{\text{'}}$ instead of M$7).$

Prove that $F$ from question $2$ is equivalent to $F=A+C^{\text{'}}$ using proof by rewrite. HINT: remember the uniting law is the key to minimization.

Draw the gate diagram for $F=A+C.$