Which of the following logic gates is considered a universal gate because it can implement any Boolean function?

Question

BlackTom AI · Accepted Answer

NAND is functionally complete (universal) because any Boolean function can be built from NAND gates alone.

1) Relevant concepts
- Boolean functions: mappings from binary inputs to binary outputs using logic operators (AND, OR, NOT, etc.).
- Functional completeness (universality): a set of gates is universal if every possible Boolean function can be expressed using only gates from that set.
- NAND gate: output = NOT (A AND B).

2) Step-by-step reasoning (constructing basic gates from NAND)
To show universality it is enough to construct a NOT and either AND or OR from NAND:
- NOT A from NAND: A NAND A = NOT (A AND A) = NOT A.
- AND from NAND: A AND B = NOT (A NAND B). Using NAND gates:
  Step 1: X = A NAND B
  Step 2: (A AND B) = X NAND X = NOT X = NOT (A NAND B) = A AND B.
- OR from NAND (using De Morgan and NAND-produced NOTs):
  A OR B = NOT (NOT A AND NOT B)
  Use NAND to make NOT A and NOT B:
    NA = A NAND A = NOT A
    NB = B NAND B = NOT B
  Then OR = NA NAND NB = NOT (NA AND NB) = A OR B.

3) Intermediate expressions shown
- NOT A = A NAND A
- A AND B = (A NAND B) NAND (A NAND B)
- A OR B = (A NAND A) NAND (B NAND B)

Those constructions demonstrate that NAND alone yields NOT, AND, OR and therefore can express any Boolean expression.

4) Why the correct answer is correct
Because NAND can produce the basic operations (NOT, AND, OR) through compositions, it is functionally complete. Any Boolean function can be written in terms of AND, OR, NOT (e.g., via sum-of-products) and thus can be implemented using only NAND gates.

5) Why the other options are incorrect
- a. AND gate: not universal by itself because AND is monotone and cannot produce negation (NOT) without additional constant inputs or other gates.
- b. OR gate: similarly not universal alone because it cannot produce NOT.
- d. XOR gate: XOR is not functionally complete by itself (it is a linear/parity operator); it cannot express arbitrary nonlinear functions like AND without additional non-linear gates.

6) Conclusion
Therefore the NAND gate is the universal gate. The correct answer is c. NAND gate.

Which of the following logic gates is considered a universal gate because it can implement any Boolean function?单项选择题

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