$(20$ Pts$)$ A binary to $7-$segment display is a device that takes a $4-$bit input, and creates $7$ outputs. Each output indicates whether a portion of a $7-$segment display should be lit or unlit.

I

$\backslash$table$[[$Symbol$,$b$3,$b$2,$b$1,$b$0,$c$],[0,0,0,0,0,],[1,0,0,0,1,],[2,0,0,1,0,],[3,0,0,1,1,],[4,0,1,0,0,],[5,0,1,0,1,],[6,0,1,1,0,],[7,0,1,1,1,],[8,1,0,0,0,],[9,1,0,0,1,],[$A$,1,0,1,0,],[$B$,1,0,1,1,],[$C$,1,1,0,0,],[$D$,1,1,0,1,],[$E$,1,1,1,0,],[$F$,1,1,1,1,]]$

Consider the segment on the lower right portion of the $7-$segment display.

a$)$ Create a truth table for the bottom segment, segment $c.$

b$)$ Create a Karnaugh map. You'll want two copies! One for SOP and other for POS.

c$)$ Circle the terms to find the simplest set of minterms $($sum of products$).$

d$)$ What is the algebraic expression for the minterms?

e$)$ Circle the terms to find the simplest set of maxterms $($product of sums$).$

$f)$ What is the algebraic expression for the maxterms?