Q$2.(20$ pts$.)$ A seven$-$segment display decoder takes a $4-$bit data input $D3-$ D$0$ and produces seven outputs to control LEDs to display a digit from $0$ to $9.$ The segments$/$outputs are labeled below and named as $S_a,S_b,$ etc.

a$.$ Write the truth table for this display.

b$.$ Using Karnaugh maps, determine the logical expression for the segment a $(S_a).$