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Idempotent law: A AND A = A, A OR A = A

The idempotent law says that combining a value with itself by AND or by OR gives the value back: A AND A = A and A OR A = A.

Engine name: IdempotenceFR: IdempotenceDE: Idempotenzgesetz

Combining a value with itself by AND or OR gives the same value.

Idempotent means that applying the operation again changes nothing. Asking twice whether A holds is the same as asking once, and requiring A and also A is the same as requiring A. Ordinary arithmetic has no such law (x + x is 2x), which is one of the visible differences between Boolean algebra and the algebra of numbers.

The simplifier uses idempotence to merge duplicate operands, so A AND B AND A becomes A AND B, and the normal-form converter uses it after a distribution has copied a term into two places. It is also behind the rule that a truth table never needs the same variable twice: A AND A has one input, not two, and the engine's diagram draws one input node with fan-out to both ports.

The proof below has two rows, one per value of A, and both sides agree on each.

Statement

A AND A = AA OR A = A

Proof by truth table: A AND A = A

A AND A = A
#Alhsrhs
0FALSEFALSEFALSE
1TRUETRUETRUE
lhs ≡ rhs holds · lhs and rhs agree on every assignment. · Exhaustively checked and solver cross-checked · 2 rows

The same law in four notations

WordsA AND A
ProgrammingA && A
MathematicalA ∧ A
AlgebraA·A
LaTeXA \cdot A
WordsA
ProgrammingA
MathematicalA
AlgebraA
LaTeXA
A AND A = A A AND A = A A lhsPRIMARY rhs AND
A AND A = A A AND A = A A lhsPRIMARY rhs AND

Proof by truth table: A OR A = A

A OR A = A
#Alhsrhs
0FALSEFALSEFALSE
1TRUETRUETRUE
lhs ≡ rhs holds · lhs and rhs agree on every assignment. · Exhaustively checked and solver cross-checked · 2 rows

The same law in four notations

WordsA OR A
ProgrammingA || A
MathematicalA ∨ A
AlgebraA + A
LaTeXA + A
WordsA
ProgrammingA
MathematicalA
AlgebraA
LaTeXA
A OR A = A A OR A = A A lhsPRIMARY rhs OR
A OR A = A A OR A = A A lhsPRIMARY rhs OR

The common slip

Expecting A OR A to be “more true” or A AND A to be stricter. Boolean values do not accumulate: there is no quantity to double.

Questions

Is XOR idempotent?

No: A XOR A is FALSE, because XOR is true only when its operands differ. That identity has its own page under the XOR identities.

Related

Lesson that teaches this

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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.