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

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
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