A logic circuit realizes a function $(,,,).$ The three inputs $,,$ and $$ are the

binary representation of the digits $0$ through $7$ with $$ being the MSB$.$ The input $$ is

an odd$-$parity bit; that is$,$ the value of $$ is such that $,,,$ and $$ always contains an

odd number of $1$s$.$ The function $$ has value $1$ if the input digit is a prime number. $($A

number is prime if it is divisible only by itself and $1$; $1$ is considered to be prime, and $0$

is not$).$ Consider output as don$$t care for invalid inputs.

$($a$)$ Draw a Karnaugh map for $.$

$($b$)$ Find all prime implicants and Essential prime implicants of $.$

$($c$)$ Find all minimum sum of products for $.$ Realize the minimum SOP using only

required numbers of $2-$input NAND gates.

$($d$)$ Find all prime implicants and essiential prime implicants of $.$

$($e$)$ Find all minimum product of sums for $.$ Realize the minimum POS using required

numbers of $2-$input NOR gates onl