A binary coded decimal $($DCB$)$ to $7-$segment decoder is a combinatorial circuit that converts a DCB digit into values suitable for selecting the correct segments and displaying the digit on a $7-$segment display. The seven decoder outputs $(a,b,c,d,e,f,g)$ select the corresponding segments of the $7-$segment display $($segment displayed $-$ output at $"1",$ segment not displayed $=$ output at $"0").$ The desired numerical display to represent each digit is given below. Using a truth table and Karnaugh tables, design the DCB to $7-$segment decoder using as few logic gates as possible. The six invalid combinations must not light up any segment $($i$.$e$.$ outputs at $0).$

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