Gate universality and functional completeness: Determine whether one can build a logical AND function using only NOR gates (i.e., express A * B solely with NORs).
Digital Electronics
Describing Logic Circuits
Difficulty: Easy
Choose an option
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ACorrect
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BIncorrect
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COnly if A = B
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DOnly with XOR assistance
Answer
Correct Answer: Correct
Explanation
Introduction / Context: NOR and NAND gates are universal, meaning any Boolean function can be implemented using only one of these gate types. Understanding equivalences via De Morgan’s theorems is a staple of digital logic design and optimizations in standard-cell libraries.Given Data / Assumptions:
- Target: implement AND using only NOR.
- Available primitive: 2-input NOR (and inverter via tying inputs).
- Boolean identities and De Morgan’s laws are applicable.
Concept / Approach: From De Morgan: A * B = (A' * B') = (A' + B')'. A NOR gate computes (X + Y)'. If we can obtain A' and B', then a single NOR of A' and B' yields (A' + B')' = A * B. Inverters can be made by tying both inputs of NOR together (X NOR X = X').Step-by-Step Solution:
Generate A' using NOR: A NOR A = A'.Generate B' using NOR: B NOR B = B'.Compute AND: (A' NOR B') = (A' + B')' = A * B.Thus, AND is realizable using only NOR gates.Verification / Alternative check:
Truth table inspection confirms the equivalence for all input pairs (00, 01, 10, 11).Why Other Options Are Wrong:
Incorrect: Conflicts with NOR universality.Only if A = B / Only with XOR assistance: Unnecessary constraints; the construction works for all inputs and requires no XOR.Common Pitfalls:
Forgetting that tying NOR inputs provides inversion.Confusing NOR-only with OR-only implementations; OR alone is not universal without inversion.Final Answer:
Correct