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Absorption law: A OR (A AND B) = A, with proof

The absorption law says that a value absorbs any term that contains it: A OR (A AND B) = A and A AND (A OR B) = A.

Engine name: AbsorptionFR: AbsorptionDE: Absorptionsgesetz

Combining a value with a term that contains it absorbs the term: the result is the value itself.

If A is true, A OR (A AND B) is true already; if A is false, A AND B is false too and the whole OR is false. So the second term never changes the outcome and can be dropped. The dual form works the same way: when A is false the AND is false, and when A is true A OR B is true, so the AND is A.

Absorption is the law that removes redundant conditions from rules. A policy that says “allow if the user is an admin, or an admin who has confirmed their email” allows exactly the admins; the second clause is absorbed. The worked example shows the simplifier applying it in one step with the equivalence check.

The proof below is the four-row table over A and B, where the columns for both sides agree on every row.

Statement

A AND (A OR B) = AA OR (A AND B) = A

Proof by truth table: A AND (A OR B) = A

A AND (A OR B) = A
#ABlhsrhs
0FALSEFALSEFALSEFALSE
1FALSETRUEFALSEFALSE
2TRUEFALSETRUETRUE
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 (A OR B)
ProgrammingA && (A || B)
MathematicalA ∧ (A ∨ B)
AlgebraA·(A + B)
LaTeXA \cdot \left(A + B\right)
WordsA
ProgrammingA
MathematicalA
AlgebraA
LaTeXA
A AND (A OR B) = A A AND (A OR B) = A A B lhsPRIMARY rhs AND OR
A AND (A OR B) = A A AND (A OR B) = A A B lhsPRIMARY rhs AND OR

Proof by truth table: A OR (A AND B) = A

A OR (A AND B) = A
#ABlhsrhs
0FALSEFALSEFALSEFALSE
1FALSETRUEFALSEFALSE
2TRUEFALSETRUETRUE
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 OR (A AND B)
ProgrammingA || (A && B)
MathematicalA ∨ (A ∧ B)
AlgebraA + (A·B)
LaTeXA + \left(A \cdot B\right)
WordsA
ProgrammingA
MathematicalA
AlgebraA
LaTeXA
A OR (A AND B) = A A OR (A AND B) = A A B lhsPRIMARY rhs OR AND
A OR (A AND B) = A A OR (A AND B) = A A B lhsPRIMARY rhs OR AND

The common slip

Confusing absorption with distribution. A OR (A AND B) simplifies to A, but A OR (C AND B) does not simplify at all; the shared variable is what makes absorption apply.

Questions

Is there an absorption law for XOR?

No. A XOR (A AND B) is not A: its disjunctive normal form, which the converter derives from XOR elimination, is A AND NOT B. Absorption needs the lattice structure of AND and OR, which XOR does not have.

Related

Worked examples that use it

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.