NOR gate: truth table for 2, 3 and 4 inputs, universal gate
NOR is an OR followed by a NOT: NOR(A, B) is TRUE only when both inputs are FALSE. Like NAND it is universal, and it is the gate behind “neither A nor B”.
The two-input table is the OR table inverted: one TRUE row, the first, where A and B are both FALSE, and three FALSE rows. For three and four inputs the language writes NOT (A OR B OR C) and NOT (A OR B OR C OR D), shown below, each TRUE on row 0 only. As with NAND, the gate is accepted only as NOR(A, B) with exactly two operands, because a chain of NOR gates is not associative and would have no agreed grouping.
By De Morgan, NOR(A, B) is NOT A AND NOT B: both inputs fail. That is the precise meaning of “neither A nor B”, and the pitfall page shows why NOT A OR NOT B (which is NAND) is the wrong translation. In a diagram the gate is the OR shape with a bubble on its output.
Universality works as for NAND, with the roles of AND and OR swapped: NOR(A, A) is NOT A, NOR(NOR(A, B), NOR(A, B)) is A OR B, and NOR(NOR(A, A), NOR(B, B)) is A AND B. The NOR-only converter applies these constructions, reports the gate count and checks the result exactly, and the comparison page sets the NAND and NOR counts for one rule side by side.
Accepted spellings
NOR(a, b)
NOR(a, b)NOR(a, b)NOR(a, b)NOR(a, b)2 inputs: NOR(A, B)
| # | A | B | f |
|---|---|---|---|
| 0 | FALSE | FALSE | TRUE |
| 1 | FALSE | TRUE | FALSE |
| 2 | TRUE | FALSE | FALSE |
| 3 | TRUE | TRUE | FALSE |
Beginnerf is true exactly when neither A nor B is true.
In every other case, f is false.
About the operators:
- NOR (“not or”) is true only when both parts are false.
Intermediatef is true exactly when neither A nor B is true.
In every other case, f is false.
Advancedf is true exactly when neither A nor B is true [Peirce arrow ↓ (NOR)].
In every other case, f is false.
Formal notation: NOR(A, B)
3 inputs: NOT (A OR B OR C): NOT (A OR B OR C)
| # | A | B | C | f |
|---|---|---|---|---|
| 0 | FALSE | FALSE | FALSE | TRUE |
| 1 | FALSE | FALSE | TRUE | FALSE |
| 2 | FALSE | TRUE | FALSE | FALSE |
| 3 | FALSE | TRUE | TRUE | FALSE |
| 4 | TRUE | FALSE | FALSE | FALSE |
| 5 | TRUE | FALSE | TRUE | FALSE |
| 6 | TRUE | TRUE | FALSE | FALSE |
| 7 | TRUE | TRUE | TRUE | FALSE |
Beginnerf is true exactly when the following condition does not hold:
1. at least one of A, B, or C is true (several may be true)
In every other case, f is false.
About the operators:
- NOT reverses a value: true becomes false and false becomes true.
- OR is inclusive: it is true when at least one part is true, including when several are.
Intermediatef is true exactly when the following condition does not hold:
1. at least one of A, B, or C is true (several may be true)
In every other case, f is false.
Advancedf is true exactly when the following condition does not hold [negation ¬]:
1. at least one of A, B, or C is true (several may be true) [inclusive disjunction ∨]
In every other case, f is false.
Formal notation: ¬(A ∨ B ∨ C)
4 inputs: NOT (A OR B OR C OR D): NOT (A OR B OR C OR D)
| # | A | B | C | D | f |
|---|---|---|---|---|---|
| 0 | FALSE | FALSE | FALSE | FALSE | TRUE |
| 1 | FALSE | FALSE | FALSE | TRUE | FALSE |
| 2 | FALSE | FALSE | TRUE | FALSE | FALSE |
| 3 | FALSE | FALSE | TRUE | TRUE | FALSE |
| 4 | FALSE | TRUE | FALSE | FALSE | FALSE |
| 5 | FALSE | TRUE | FALSE | TRUE | FALSE |
| 6 | FALSE | TRUE | TRUE | FALSE | FALSE |
| 7 | FALSE | TRUE | TRUE | TRUE | FALSE |
| 8 | TRUE | FALSE | FALSE | FALSE | FALSE |
| 9 | TRUE | FALSE | FALSE | TRUE | FALSE |
| 10 | TRUE | FALSE | TRUE | FALSE | FALSE |
| 11 | TRUE | FALSE | TRUE | TRUE | FALSE |
| 12 | TRUE | TRUE | FALSE | FALSE | FALSE |
| 13 | TRUE | TRUE | FALSE | TRUE | FALSE |
| 14 | TRUE | TRUE | TRUE | FALSE | FALSE |
| 15 | TRUE | TRUE | TRUE | TRUE | FALSE |
Beginnerf is true exactly when the following condition does not hold:
1. at least one of A, B, C, or D is true (several may be true)
In every other case, f is false.
About the operators:
- NOT reverses a value: true becomes false and false becomes true.
- OR is inclusive: it is true when at least one part is true, including when several are.
Intermediatef is true exactly when the following condition does not hold:
1. at least one of A, B, C, or D is true (several may be true)
In every other case, f is false.
Advancedf is true exactly when the following condition does not hold [negation ¬]:
1. at least one of A, B, C, or D is true (several may be true) [inclusive disjunction ∨]
In every other case, f is false.
Formal notation: ¬(A ∨ B ∨ C ∨ D)
The common slip
Translating “neither A nor B” as NOT A OR NOT B. That formula is also TRUE when only one of them fails. “Neither” requires both to fail: NOT A AND NOT B, which is NOR(A, B), TRUE on a single row.
Questions
Which is cheaper, NAND or NOR?
It depends on the function and the technology. In the NOR family OR costs two gates and AND three; in the NAND family it is the other way round. The converter reports the count for any expression, and the comparison page discusses the choice.
What is NOR with a constant input?
NOR(A, FALSE) is NOT A, the constant-input inverter the converter uses when duplicating a signal would be too large, and NOR(A, TRUE) is always FALSE, because TRUE dominates the OR inside.
Related
Guides
Worked examples that use it
Laws
- NOR elimination: NOR(A, B) = NOT (A OR B)
- 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))
Tools
- Truth table generator: any Boolean expression, up to 12 variables, with intermediate columns
- Logic circuit generator: Boolean expression to gate diagram (SVG)
Practice
See also
- NOR elimination: NOR(A, B) = NOT (A OR B)
- 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))
- De Morgan's laws: NOT (A AND B) = NOT A OR NOT B, with proof and examples
- NAND gate: truth table for 2, 3 and 4 inputs, universal gate
- “Neither A nor B” is NOT A AND NOT B
- AND with NOR gates only
- NAND-only and NOR-only converter with equivalence proof
- NAND vs NOR as the universal gate
Operators and gates
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