Question

A system has 4 input lines {A,B,C,D} and 1 output line {F}. The input lines {A, B, C} of the system are connected 

to the selection lines {S2 , S1 , S0} of an 8×1 multiplexer, respectively. The connection configuration of data 

input lines {i0, i1, … , i7 } of the multiplexer are given as follows: 

i1= i2 = D; i3 = i5 =1; i0 =i6 = D' ; and i7 = i4 = 0 

Determine the Boolean function that the output of the multiplexer implements. Also, provide a minimized SoP 

expression for the obtained Boolean function F(A,B,C,D).

Answer 1

(a) Here GT A,B,C, = $S_2,S_1,S_0$ respectively

G.T $I_1=I_2=D$ $I_3=I_5=D$ $I_7=I_4=0$

$F(A,B,C,D)=\bar{A}\bar{B}CD+\bar{A}B\bar{C}D+\bar{A}BC\bar{D}+A\bar{B}C\bar{D}$

Minimized = $\sum f (3,5,6,12)$

sop form

f(A,,B,C,D)

(b) $I_1=I_4=D$ $I_3=I_5=1$ $I_0=I_6=B_0$

$I_7=I_2=D$

substitute in the above expression ,

F(A,B,C,D)= $\bar{A}\bar{B}CD+\bar{A}\bar{B}\bar{C}\bar{D}+\bar{A}BC+A\bar{B}\bar{C}+A\bar{B}\bar{C}\bar{D}+A\bar{B}C+ABC\bar{D}$

Y=$\bar{S_2,S_1,S_0} I_0+S_2S_1S_0I_1+S_2S_1S_0I_2+S_2S_1S_0I_3$

$S_2S_1S_0I_4+S_2S_1S_0I_5+S_2S_1S_0I_6+S_2S_1S_0I_7$

$F(A,B,C,D)=\bar{A}\bar{B}CD+\bar{A}\bar{B}\bar{C}\bar{D}+\bar{A}BC+A\bar{B}\bar{C}+A\bar{B}\bar{C}\bar{D}+A\bar{B}C+ABC\bar{D}$

$\bar{A}BC=\bar{A}BC(D+\bar{D})$ as $D+\bar{D}=1$

$F(A,B,C,D)=\sum(3,4,6,7,8,10,11,13)$

minimal pos expression

$\pi(0,1,2,5,9,12,14,15)$

(c) F(A,B,C,D)= CD+ABCD+ABC+ABCD

$I$ is connected to $I_7$ D s connected to $I_1$

D is connected to $I_5,I_0,I_2,I_4,1_6$ 0 a connected to $I_3$