Question

Q.3-A). The circuit of Figure below shows how an eight-input MUX can be used to generate a four-variable logic function, even though the MUX has only three SELECT inputs. (a) Set up a truth table showing the output Z for the 16 possible combinations of input variables. b) Write the sum-of-products expression for Z and simplify it. 45 V 74HC151 MUX

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Answer #1

Step 1:

(a) Set up a truth table showing the output Z for the 16 possible combinations of the input variable

First set up a truth table for C, B, A selection line and Z output using given diagram in the problem:

Truth Table:1

Selection Line(Input) Output
S2 S1 S0 Z
C B A
0 0 0 0
0 0 1 D
0 1 0 1
0 1 1 0
1 0 0 php8xF3RP.png
1 0 1 0
1 1 0 0
1 1 1 0

Step 2:

Now we set up a truth table for all 4 input (A, B, C, D)and one output Z

Using Truth Table:1

the Z output for all 16 possible combinations

Truth Table:2

Input Selection Line(Input) Output
D S2 S1 S0 Z
C B A
0 0 0 0 0
0 0 0 1 0
0 0 1 0 1
0 0 1 1 0
0 1 0 0 1
0 1 0 1 0
0 1 1 0 0
0 1 1 1 0
1 0 0 0 0
1 0 0 1 1
1 0 1 0 1
1 0 1 1 0
1 1 0 0 0
1 1 0 1 0
1 1 1 0 0
1 1 1 1 0

Step 3:

.(b) Now the Sum of Product expression

From Truth Table 2, the Minterm is:

Input Selection Line(Input) Output Minterm
D S2 S1 S0 Z
C B A
0 0 0 0 0 0
0 0 0 1 0 0
0 0 1 0 1 DCBA
0 0 1 1 0 0
0 1 0 0 1 DcBA
0 1 0 1 0 0
0 1 1 0 0 0
0 1 1 1 0 0
1 0 0 0 0 0
1 0 0 1 1 DCBA
1 0 1 0 1 DCBĀ
1 0 1 1 0 0
1 1 0 0 0 0
1 1 0 1 0 0
1 1 1 0 0 0
1 1 1 1 0 0

the Sum of Product expression is:

Z = DCBĀ + DCBĀ+DCB A+ DCBĀ

Z = DCBĀ + DCBĀ + ŪCĒĀ + DCBA

Simplify the above expression

Z = CBĀCD + D) + ĐCĐĀ+ DCZA

We know that : D+D = 1

Z = CBĀ + DCBA+DCBA

Z = ĀBC + ABCD + AB CD

So that Sum of Product expression after simplification:

Z = ĀBC + ABCD + AB CD

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