Functional completeness in Boolean logic: Which of the following gate sets is NOT functionally complete for constructing arbitrary Boolean functions?

Introduction / Context:
A set of logic primitives is called functionally complete if any Boolean function can be constructed using only gates from that set. Recognizing complete versus incomplete sets is critical for hardware minimization and theoretical foundations of digital logic design.

Given Data / Assumptions:

• NAND alone is known to be functionally complete.
• NOR alone is also functionally complete.
• AND and OR together, without inversion, cannot generate complements.

Concept / Approach:
To synthesize arbitrary Boolean functions, the gate set must be able to express AND, OR, and NOT operations (directly or indirectly). NAND and NOR each can create NOT and, with combinations, produce both AND and OR. However, the pair AND + OR lacks intrinsic inversion; without NOT, it cannot create functions that require complementing variables.

Step-by-Step Solution:
List candidate sets and check whether NOT can be synthesized.Verify that NAND can build NOT, AND, and OR; similarly for NOR.Conclude that “AND, OR” is not functionally complete because it cannot produce NOT by itself.

Verification / Alternative check:
Standard proofs show NOT(A) = NAND(A, A) or NOT(A) = NOR(A, A). No such identity exists using only AND and OR without a complemented input available.

Why Other Options Are Wrong:
NAND and NOR each form complete bases. AND, OR, NOT is trivially complete by definition.

Common Pitfalls:
Assuming complemented literals are “given” externally; functional completeness is judged without external inversion sources.

Final Answer:
AND, OR.