##  [Commutative Law](/commutative-law-0) 

 Definition

The principle in logic and algebra that a binary operator yields the same result when its operands are permuted; formally, for operands A and B and a commutative connective ⊗, A ⊗ B = B ⊗ A (examples: A ∧ B = B ∧ A, A ∨ B = B ∨ A).

 

 

 

 

 

 





## Principle

Principle

Order of operands does not matter for the operator: permutation of inputs leaves the operation invariant.

 

 

 

 

 





## Demonstration

Demonstration

In Boolean conjunction, A ∧ B equals B ∧ A because both evaluate to true exactly when A and B are true; truth tables for both orders are identical.

 

 

 

 

## Misapplication

Misapplication

Assuming commutativity for implication (A → B ≠ B → A in general) or for noncommutative operations such as Boolean difference, function composition, or matrix multiplication leads to incorrect transformations.

 

 

 

 

 





## Consequence

Consequence

Expressions can be rearranged to simplify formulas, factor common subexpressions, or choose canonical operand order for normalization and optimization.

 

 

 

 

## Reversal

Reversal

Non-commutativity: operators whose results change under permutation of operands (e.g., subtraction a − b, implication A → B, ordered pair constructors).

 

 

 

 

 





## Boundary

Boundary

Applies only to the specified binary operator and its algebraic context; a binary operator may be commutative in one algebra and not in another, and unary operators are outside its scope.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension with ordered or directional connectives—commutativity conflicts with concepts that encode direction, causality, or sequence (e.g., implication versus conjunction).

 

 

 

 

 





## Synthesis

Synthesis

Commutative law identifies when operand order is immaterial for a given binary operation, enabling rearrangement and canonicalization where permitted while remaining distinct from directional or ordered operations.