P$3(20$ points$)$: $1-$to$-8$ demultiplexer

a$)$ Write the Boolean functions for the outputs of a $1-$to$-8$ demultiplexer that an input $F$ with select lines $\{A,B,C\}.$

b$)$ Implement the demultiplexer using AND, and NOT gates.

c$)$ Implement the demultiplexer using $1-$to$-2$ demultiplexers and a minimal number of additional gates

d$)$ Implement the multiplexer using $1-$to$-4$ demultiplexers and a minimal number of additional gates