Constructing a 2-input AND using only NOR gates: Evaluate the statement: “A minimum of three universal NOR gates is required to realize the logical operation of a 2-input AND gate.”
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ACorrect
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BIncorrect
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COnly two NOR gates are sufficient for AND
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DNot enough information is provided
Answer
Correct Answer: Correct
Explanation
Introduction / Context:NOR is a universal gate, meaning any logic function can be implemented using only NOR gates. Determining the minimum number of NOR gates to realize other primitives is a common design skill, useful when standard cell availability is constrained or for theoretical proofs.
Given Data / Assumptions:
- Target function: 2-input AND, X = A * B.
- Available primitive: 2-input NOR gate.
- Standard positive-logic conventions.
Concept / Approach:Use De Morgan’s law: A * B = ~(~(A * B)) = ~(~A + ~B). Realize in two stages: first produce ~A and ~B using self-NOR (inversion), then NOR those results to obtain A * B. This construction uses three NOR gates total.
Step-by-Step Solution:
Inversion via NOR: ~A = A NOR A; ~B = B NOR B.AND via NOR: X = (~A) NOR (~B) = ~(~A + ~B) = A * B.Gate count: two inverters (2 NORs) + one combining NOR = 3 NOR gates.Verification / Alternative check:Truth table check confirms the final output matches AND. Known minimal implementations show three NORs are necessary and sufficient; two NORs cannot generate both required inversions and the final combination simultaneously.
Why Other Options Are Wrong:
Incorrect: Conflicts with standard minimal construction.Only two NORs are sufficient: With two NORs you can invert one input and combine, but you cannot invert both inputs and combine correctly for all input cases.Not enough information: Function and gate type provide everything needed.Common Pitfalls:Forgetting that NOR-inverter uses both inputs tied together; attempting to reuse intermediate nodes incorrectly; mixing NAND and NOR equivalences.
Final Answer:Correct