A Boolean digital circuit is composed using two 4-input multiplexers (M1 and…
2023
A Boolean digital circuit is composed using two 4-input multiplexers (M1 and M2) and one 2-input multiplexer (M3) as shown in the figure. X0–X7 are the inputs of the multiplexers M1 and M2 and could be connected to either 0 or 1. The select lines of the multiplexers are connected to Boolean variables A, B and C as shown.
Which one of the following set of values of (X0, X1, X2, X3, X4, X5, X6, X7) will realise the Boolean function
- A.
(1, 1, 0, 0, 1, 1, 1, 0)
- B.
(1, 1, 0, 0, 1, 1, 0, 1)
- C.
(1, 1, 0, 1, 1, 1, 0, 0)
- D.
(0, 0, 1, 1, 0, 1, 1, 1)
Attempted by 78 students.
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Correct answer: C
Key idea: determine which input combinations of A, B, C produce output 1, map those combinations to the corresponding X0..X7 inputs, and then read off the required tuple.
How the multiplexers map inputs to X-entries:
M3 selects between the outputs of the top multiplexer (used when B = 0) and the bottom multiplexer (used when B = 1).
For each 4-input multiplexer, the two select lines are A and C with the usual ordering: (A,C) = (0,0) → input 0, (0,1) → input 1, (1,0) → input 2, (1,1) → input 3.
Therefore the full mapping X0..X7 corresponds to:
X0: B=0, A=0, C=0
X1: B=0, A=0, C=1
X2: B=0, A=1, C=0
X3: B=0, A=1, C=1
X4: B=1, A=0, C=0
X5: B=1, A=0, C=1
X6: B=1, A=1, C=0
X7: B=1, A=1, C=1
Simplify the given Boolean function:
F = ¬A + ¬A·¬C + A·¬B·C = ¬A + A·(¬B·C) = ¬A + (¬B·C).
Determine required outputs for each A value:
If A = 0 then ¬A = 1, so F = 1 for all B and C. Thus inputs with A = 0 (X0, X1, X4, X5) must be 1.
If A = 1 then F = ¬B·C, so F = 1 only when B = 0 and C = 1. That combination corresponds to X3 (B=0,A=1,C=1), so X3 = 1. All other A=1 cases (X2, X6, X7) must be 0.
Collecting these values gives the required input tuple (X0,X1,X2,X3,X4,X5,X6,X7) = (1, 1, 0, 1, 1, 1, 0, 0), which matches the correct answer.