Question $1$

$1.$a Draw an empty template of a $4-$variable Karnaugh Map, ensuring that you label the axes correctly.

$[3$ marks$]$

$1.$b i$)$ Use your Karnaugh Map template to simplify the following Boolean expression:

$F=A^{\text{'}}B{C}^{\text{'}}D+AB{C}^{\text{'}}D+$ABCD$+A{B}^{\text{'}}CD$

$($note: X$\text{'}$ means "not X$")$

Show all working and give your answer as a $($simplified$)$ sum$-$of$-$products Boolean expression.

$[4$ marks$]$

ii$)$ Using de Morgan's law, manipulate your simplified sum$-$of$-$products expression from $1.$b$.$i to generate an equivalent product$-$of$-$products expression that is ready to map to a logic circuit implementation that uses only NAND gates.

$[4$ marks$]$

iii$)$ Finally, draw the corresponding circuit using $3-$input NAND gates.

$[4$ marks$]$

$1.$c i$)$ Write down the truth table for the XOR function in terms of two inputs, A and $B.$

$[2$ marks$]$

ii$)$ Write down the corresponding sum$-$of$-$products Boolean function.

$[2$ marks$]$

iii$)$ Manipulate your Boolean function to generate an equivalent sum$-$of$-$sums expression that is ready to map to a logic circuit implementation that uses only NOR gates.

$[3$ marks$]$

iv$)$ Finally, draw the corresponding circuit using $2-$input NOR gates.

$[3$ marks$]$