OR from NOR gates: A OR B = NOR(NOR(A, B), NOR(A, B))
OR from NOR gates: A OR B = NOR(NOR(A, B), NOR(A, B)), a NOR followed by a NOR inverter.
OR is built as the negation (a NOR with tied inputs) of a NOR.
A NOR is an OR with its output inverted; inverting again restores the OR. The inverter is the tied NOR of the previous law, so the construction uses two gates, with the inner NOR appearing twice in the text because the text cannot show that its single output feeds both inputs of the second gate.
The construction is the exact dual of AND from NAND, and the engine proves it the same way, on the four rows over A and B. The NOR-only converter reports gate counts on this basis, and the comparison page sets the NAND and NOR counts for one rule side by side.
The proof is below.
Statement
A OR B = NOR(NOR(A, B), NOR(A, B)) Proof by truth table: A OR B = NOR(NOR(A, B), NOR(A, B))
| # | A | B | lhs | rhs |
|---|---|---|---|---|
| 0 | FALSE | FALSE | FALSE | FALSE |
| 1 | FALSE | TRUE | TRUE | TRUE |
| 2 | TRUE | FALSE | TRUE | TRUE |
| 3 | TRUE | TRUE | TRUE | TRUE |
The same law in four notations
A OR BA || BA ∨ BA + BA + BNOR(NOR(A, B), NOR(A, B))NOR(NOR(A, B), NOR(A, B))NOR(NOR(A, B), NOR(A, B))NOR(NOR(A, B), NOR(A, B))\overline{\overline{A + B} + \overline{A + B}}The common slip
Expecting one NOR to be an OR. A single NOR is the negation of the OR; the second gate is the inverter that restores it.
Questions
Is the NOR family used in practice?
Yes; which gate is cheaper depends on the technology. CMOS NAND gates are usually faster than NOR gates of the same size, which is one reason NAND is the more common choice, but both families appear in real designs.
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
- NOT from NOR: NOT A = NOR(A, A)
- AND from NOR gates: A AND B = NOR(NOR(A, A), NOR(B, B))
- AND with NOR gates only
- NAND vs NOR as the universal gate
Laws
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