$($d$)$ Given the following truth table for a four$-$input logic circuit:

$\backslash$table$[[$x$,$y$,$z$,$F$\_($x$,$y$,$x$)],[0,0,0,0],[0,0,1,0],[0,1,0,0],[0,1,1,1],[1,0,0,0],[1,0,1,1],[1,1,0,1],[1,1,1,1]]$

i$.$ Write the sum$-$of$-$products expression for this truth table.

$[2$ marks$]$

ii$.$ Draw the circuit for the boolean expression $f(x,y,z).$

iii. Translate the truth table into its corresponding Karnaugh map.

$[2$ marks$]$

iv$.$ Use the Karnaugh map in $($iii$)$ to minimise the sum$-$of$-$products expression in $($i$).$

$[2$ marks$]$

v$.$ Use the results in $($iv$)$ and give a more simplified circuit for $\frac{?}{(x,y,z)}.$

$[2$ marks$]$