Q$2$:

A BCD$-$to$-$seven$-$segment decoder is a combinational circuit that converts a decimal digit in BCD to an appropriate code for the selection of segments in an indicator used to display the decimal digit in a familiar form. The seven outputs of the decoder $,d,e,f,g$ select the corresponding segments in the display, as shown in Fig. P$4\mathrm{.}9($a$).$ The numeric display chosen to represent the decimal digit is shown in Fig. P$4\mathrm{.}9($b$).$ Using a truth table and Karnaugh maps, design the BCD$-$to$-$seven$-$segment decoder using a minimum number of gates. The six invalid combinations should result in a blank display.