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Null law (domination): A AND FALSE = FALSE, A OR TRUE = TRUE

The null law (the engine calls it domination) says that the dominant constant fixes the result whatever the other operand is: FALSE dominates AND and TRUE dominates OR.

Engine name: DominationFR: Élément absorbantDE: Dominanzgesetz

The dominant constant (FALSE for AND, TRUE for OR) fixes the result whatever the other operand is.

A AND FALSE is false no matter what A is, because an AND needs every operand true. A OR TRUE is true no matter what A is, because an OR needs only one. In algebra notation the law reads A·0 = 0 and A + 1 = 1, and the name null law comes from the first form, where the zero annihilates the product.

This is the law that lets a whole subtree disappear. When a complement law produces FALSE inside an AND, or TRUE inside an OR, the null law removes everything else at that level, which is often the largest single reduction in a simplification. The lesson on simplification opens with identity and domination.

In hardware the law says that an AND gate with one input stuck low is stuck low, which is why a stuck-at fault on one input can hide the behaviour of all the others.

Statement

A AND FALSE = FALSEA OR TRUE = TRUE

Proof by truth table: A AND FALSE = FALSE

A AND FALSE = 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 FALSE
ProgrammingA && FALSE
MathematicalA ∧ 0
AlgebraA·0
LaTeXA \cdot 0
WordsFALSE
ProgrammingFALSE
Mathematical0
Algebra0
LaTeX0
A AND FALSE = FALSE A AND FALSE = FALSE A lhsPRIMARY rhs AND FALSE FALSE
A AND FALSE = FALSE A AND FALSE = FALSE A lhsPRIMARY rhs AND FALSE FALSE

Proof by truth table: A OR TRUE = TRUE

A OR TRUE = 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 TRUE
ProgrammingA || TRUE
MathematicalA ∨ 1
AlgebraA + 1
LaTeXA + 1
WordsTRUE
ProgrammingTRUE
Mathematical1
Algebra1
LaTeX1
A OR TRUE = TRUE A OR TRUE = TRUE A lhsPRIMARY rhs OR TRUE TRUE
A OR TRUE = TRUE A OR TRUE = TRUE A lhsPRIMARY rhs OR TRUE TRUE

The common slip

Applying the null law to the wrong constant. A AND TRUE keeps A (identity law); only A AND FALSE collapses to FALSE.

Questions

Why does the engine call it domination rather than null law?

Both names are in use. Textbooks on digital design tend to say null law or annulment law; logic texts say domination. The engine's catalogue uses domination; the page title carries the textbook name first so either search lands here.

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.