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AND from NOR gates: A AND B = NOR(NOR(A, A), NOR(B, B))

AND from NOR gates: A AND B = NOR(NOR(A, A), NOR(B, B)), a NOR of the two inverted inputs.

Engine name: AND as NORFR: AND en NORDE: AND als NOR

AND is built as a NOR of the two negated inputs, each negation being a NOR with tied inputs.

By De Morgan, A AND B is NOT (NOT A OR NOT B), which is a NOR of NOT A and NOT B; with each inverter a tied NOR, the AND costs three gates. This completes the NOR family: NOT, OR and AND are all available from NOR alone, so any Boolean function can be built from it.

The worked example on AND with NOR gates proves the construction and shows its diagram, and the NOR-only converter applies it to whole expressions with the gate count and the equivalence check.

The four-row proof is below.

Statement

A AND B = NOR(NOR(A, A), NOR(B, B))

Proof by truth table: A AND B = NOR(NOR(A, A), NOR(B, B))

A AND B = NOR(NOR(A, A), NOR(B, B))
#ABlhsrhs
0FALSEFALSEFALSEFALSE
1FALSETRUEFALSEFALSE
2TRUEFALSEFALSEFALSE
3TRUETRUETRUETRUE
lhs ≡ rhs holds · lhs and rhs agree on every assignment. · Exhaustively checked and solver cross-checked · 4 rows

The same law in four notations

WordsA AND B
ProgrammingA && B
MathematicalA ∧ B
AlgebraA·B
LaTeXA \cdot B
WordsNOR(NOR(A, A), NOR(B, B))
ProgrammingNOR(NOR(A, A), NOR(B, B))
MathematicalNOR(NOR(A, A), NOR(B, B))
AlgebraNOR(NOR(A, A), NOR(B, B))
LaTeX\overline{\overline{A + A} + \overline{B + B}}
A AND B = NOR(NOR(A, A), NOR(B, B)) A AND B = NOR(NOR(A, A), NOR(B, B)) A B lhsPRIMARY rhs AND NOR NOR NOR
A AND B = NOR(NOR(A, A), NOR(B, B)) A AND B = NOR(NOR(A, A), NOR(B, B)) A B lhsPRIMARY rhs AND NOR NOR NOR

The common slip

Confusing this with NOR itself. NOR(A, B) is true only when both inputs are false; the inverted inputs turn it into an AND, which is true only when both are true.

Questions

Why three gates for AND but two for OR in the NOR family?

Because NOR is OR-based: inverting a NOR gives OR directly (two gates), while AND needs De Morgan with both inputs inverted first (three gates). The NAND family is the mirror image: AND costs two, OR costs three.

Related

Lesson that teaches this

Tools

Practice

See also

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.