A combinational circuit is used to control a seven$-$segment display of decimal digits, as shown in

below figures. The circuit has four inputs, which provide the four$-$bit code used in packed

decimal representation $(010=0000,$dots,$910=1001).$ The seven outputs define which segments

will be activated to display a given decimal digit. For example, if the input is $x1=0,x2=0,$

$x3=0,x4=0,$ the output has to $Z|=1,Z2=1,Z3=1,Z4=0,Z5=1,Z6=1,Z7=1.$ Note that

some combinations of inputs and outputs are not needed.

a$.$ Develop a truth table for this circuit.

b$.$ Express the truth table in SOP form.

c$.$ Provide a simplified expression.

$($a$)$

$($b$)$