Identity law of Boolean algebra: A AND TRUE = A, A OR FALSE = A
The identity law says that combining a value with the neutral constant leaves it unchanged: TRUE is neutral for AND and FALSE is neutral for OR.
Combining a value with the neutral constant (TRUE for AND, FALSE for OR) leaves it unchanged.
A AND TRUE can only be true when A is, and is false exactly when A is false, so it is A. A OR FALSE is true exactly when A is true, so it is A as well. The neutral element plays the role that 1 plays for multiplication and 0 for addition, which is why the algebra notation writes the law as A·1 = A and A + 0 = A.
The simplifier uses this law constantly, usually after another law has produced a constant: a complement law turns B OR NOT B into TRUE, and the identity law then removes the TRUE from A AND (B OR NOT B). In circuits, an AND gate with one input tied high is a wire, and an OR gate with one input tied low is a wire; the law is why such gates can be deleted.
The proof is the truth table below: with A as the only variable there are two rows, and both sides agree on each.
Statement
A AND TRUE = AA OR FALSE = A Proof by truth table: A AND TRUE = A
| # | A | lhs | rhs |
|---|---|---|---|
| 0 | FALSE | FALSE | FALSE |
| 1 | TRUE | TRUE | TRUE |
The same law in four notations
A AND TRUEA && TRUEA ∧ 1A·1A \cdot 1AAAAAProof by truth table: A OR FALSE = A
| # | A | lhs | rhs |
|---|---|---|---|
| 0 | FALSE | FALSE | FALSE |
| 1 | TRUE | TRUE | TRUE |
The same law in four notations
A OR FALSEA || FALSEA ∨ 0A + 0A + 0AAAAAThe common slip
Mixing up which constant is neutral for which operator. A OR TRUE is not A; it is TRUE (that is the null law, called domination here). The neutral constant is the one that cannot decide the result on its own.
Questions
Why is the law written with 1 and 0 in algebra notation?
Boolean algebra borrowed the notation of ordinary algebra: AND is written as a product and OR as a sum, so TRUE and FALSE become 1 and 0. The engine prints the same law in all four notations, and the parser accepts any of them.
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
- Boolean simplification exercises with answers: basic
- Boolean simplification exercises with answers: intermediate
- Boolean simplification exercises with answers: advanced
See also
- Null law (domination): A AND FALSE = FALSE, A OR TRUE = TRUE
- Lesson 7: Simplification laws (intermediate, 12 min)
- Absorption in practice: A OR (A AND B) simplified
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.