NOR elimination: NOR(A, B) = NOT (A OR B)
NOR elimination rewrites NOR(A, B) as NOT (A OR B): a NOR is an OR followed by a NOT, true only when both inputs are false.
NOR is rewritten as the negation of an OR.
NOR is the “neither nor” gate: NOR(A, B) holds exactly when neither A nor B does, which by De Morgan is NOT A AND NOT B. The elimination law is the definition and is the named step the engine takes before converting an expression with NOR gates to a normal form or to another gate basis.
Like NAND, NOR is universal, and the NOR construction laws (NOT, OR and AND from NOR alone) are proved on their own pages. The comparison page counts the gates each basis needs for the same rule, which is the practical question when one family is cheaper or faster than the other.
The four-row proof is below.
Statement
NOR(A, B) = NOT (A OR B) Proof by truth table: NOR(A, B) = NOT (A OR B)
| # | A | B | lhs | rhs |
|---|---|---|---|---|
| 0 | FALSE | FALSE | TRUE | TRUE |
| 1 | FALSE | TRUE | FALSE | FALSE |
| 2 | TRUE | FALSE | FALSE | FALSE |
| 3 | TRUE | TRUE | FALSE | FALSE |
The same law in four notations
NOR(A, B)NOR(A, B)NOR(A, B)NOR(A, B)\overline{A + B}NOT (A OR B)!(A || B)¬(A ∨ B)(A + B)'\overline{A + B}The common slip
Reading “neither A nor B” as NOT A OR NOT B. That is NAND, which is also true when only one of them fails. “Neither” requires both to fail: NOT A AND NOT B, which is NOR.
Questions
Why does the engine accept only NOR(A, B) with two operands?
For the same reason as NAND: a chain of NORs is not associative, so an unbracketed infix chain would be ambiguous. A three-input NOR is written NOT (A OR B OR C), which the operator page shows.
Related
Lesson that teaches this
Pitfalls
Tools
- Boolean algebra calculator and simplifier with steps (every law named and verified)
- Logical equivalence checker: two expressions, exact verdict, counterexample
See also
- NOR gate: truth table for 2, 3 and 4 inputs, universal gate
- NOT from NOR: NOT A = NOR(A, A)
- 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))
- “Neither A nor B” is NOT A AND NOT B
Laws
Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build . Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.