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Complement law: A AND NOT A = FALSE, A OR NOT A = TRUE

The complement law says that a value combined with its own negation gives FALSE under AND and TRUE under OR: A AND NOT A = FALSE, A OR NOT A = TRUE.

Engine name: ComplementFR: ComplémentaritéDE: Komplementärgesetz

A value combined with its own negation gives FALSE under AND and TRUE under OR.

Nothing can be both true and false, so A AND NOT A never holds: it is the simplest contradiction. Everything is either true or false, so A OR NOT A always holds: it is the simplest tautology, the law of the excluded middle. The two forms are each other's dual, and De Morgan's laws turn one into the other.

In simplification the complement law is the usual source of constants, which the identity and null laws then remove. In testing it explains vacuous results: a rule whose assumptions include both A and NOT A has no admissible case, so every property holds of it for a bad reason, and the engine labels such results vacuous rather than passed.

The two-row proof below covers both values of A.

Statement

A AND NOT A = FALSEA OR NOT A = TRUE

Proof by truth table: A AND NOT A = FALSE

A AND NOT A = FALSE
#Alhsrhs
0FALSEFALSEFALSE
1TRUEFALSEFALSE
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 NOT A
ProgrammingA && !A
MathematicalA ∧ ¬A
AlgebraA·A'
LaTeXA \cdot \overline{A}
WordsFALSE
ProgrammingFALSE
Mathematical0
Algebra0
LaTeX0
A AND NOT A = FALSE A AND NOT A = FALSE A lhsPRIMARY rhs AND NOT FALSE
A AND NOT A = FALSE A AND NOT A = FALSE A lhsPRIMARY rhs AND NOT FALSE

Proof by truth table: A OR NOT A = TRUE

A OR NOT A = TRUE
#Alhsrhs
0FALSETRUETRUE
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 NOT A
ProgrammingA || !A
MathematicalA ∨ ¬A
AlgebraA + A'
LaTeXA + \overline{A}
WordsTRUE
ProgrammingTRUE
Mathematical1
Algebra1
LaTeX1
A OR NOT A = TRUE A OR NOT A = TRUE A lhsPRIMARY rhs OR NOT TRUE
A OR NOT A = TRUE A OR NOT A = TRUE A lhsPRIMARY rhs OR NOT TRUE

The common slip

Reading A OR NOT A as an exclusive or (“A or not A, but not both”). It is an inclusive OR that happens to be always true; the exclusive version, A XOR NOT A, is also always true but for a different reason, and the engine proves both.

Questions

Is this the same as the law of the excluded middle?

The OR form is exactly the law of the excluded middle, and the AND form is the law of non-contradiction. Boolean algebra states them as one pair because each follows from the other by De Morgan.

Related

Worked examples that use it

Lesson that teaches this

Pitfalls

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.