Problem $3$:

$(1)$Use K$-$maps to find MSOP expressions for the following incompletely specified functions, and draw the logic circuit.

$f(a,b,c,d)={\displaystyle \underset{?}{\overset{?}{}}}m(1,5,7,9)+d(6,13)$

$(2)$ Determine which of the following functions are equivalent.

$f_1(A,B,C,D)=AC+BD+\frac{A}{b}ar\frac{B}{b}ar(D)$

$f_2(A,B,C,D)=\frac{A}{b}ar\frac{B}{b}ar(D)+AB\frac{+}{b}ar(A)\frac{B}{b}ar(C)$

$f_3(A,B,C,D)=BD+\frac{A}{b}ar\frac{B}{b}ar(D)+$ACD$+$ABC

$f_4(A,B,C,D)=AC+\frac{A}{b}ar\frac{B}{b}ar(CD)\frac{+}{b}ar(A)BD+\frac{B}{b}ar(C)D$

$f_5(A,B,C,D)=(B\frac{+}{b}ar(D))(A+B)(A\frac{+}{b}ar(C))$