NOT from NOR: NOT A = NOR(A, A)
NOT from NOR: an inverter is a NOR gate with both inputs tied together, NOT A = NOR(A, A).
NOT is built from one NOR gate whose two inputs receive the same value.
NOR(A, A) is NOT (A OR A), and A OR A is A by idempotence, so the gate outputs NOT A. It is the NOR family's first construction, the mirror image of the NAND inverter, and the NOR-only converter uses it by default, with NOR(A, FALSE) as the constant-input alternative when duplicating a signal would be too costly.
With an inverter available, the NOR family builds OR as a NOR followed by this inverter and AND as a NOR of two inverted inputs, completing the proof that NOR alone is universal.
The two-row proof is below.
Statement
NOT A = NOR(A, A) Proof by truth table: NOT A = NOR(A, A)
| # | A | lhs | rhs |
|---|---|---|---|
| 0 | FALSE | TRUE | TRUE |
| 1 | TRUE | FALSE | FALSE |
The same law in four notations
NOT A!A¬AA'\overline{A}NOR(A, A)NOR(A, A)NOR(A, A)NOR(A, A)\overline{A + A}The common slip
Tying the spare input of a NOR inverter high. NOR(A, TRUE) is always FALSE (TRUE dominates the OR); the constant for a NOR inverter must be FALSE.
Questions
Which constant goes with which gate?
A NAND inverter can use TRUE as its spare input (NAND(A, TRUE) = NOT A) and a NOR inverter must use FALSE (NOR(A, FALSE) = NOT A); each uses the neutral constant of the gate's underlying operator.
Related
Lesson that teaches this
Tools
- Boolean algebra calculator and simplifier with steps (every law named and verified)
- Logical equivalence checker: two expressions, exact verdict, counterexample
Practice
See also
- OR from NOR gates: A OR B = NOR(NOR(A, B), NOR(A, B))
- AND from NOR gates: A AND B = NOR(NOR(A, A), NOR(B, B))
- NOR elimination: NOR(A, B) = NOT (A OR B)
- NAND-only and NOR-only converter with equivalence proof
Laws
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