Question #194391

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).

Expert's answer

(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"

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