Commutativity¶

Definition¶

A binary operation $\star$ on $A$ is commutative if

for all $a,b\in A$$A$.

Interpretation¶

Commutativity says that the order of the two inputs does not matter: swapping $a$ and $b$ gives the same result.

Commutativity is about order, associativity about grouping, and neither implies the other. Subtraction has neither property. Composition of functions is associative but not commutative: for $f(x)=x+1$ and $g(x)=2x$$2x$, $(g\circ f)(x)=2x+2$ while $(f\circ g)(x)=2x+1$$1$. The average $a\star b=\tfrac{1}{2}(a+b)$ is commutative but not associative: $(0\star0)\star4=2$$2$, while $0\star(0\star4)=1$$1$.

Commutativity is the condition that turns a group into an Abelian group, and both operations of a field are commutative. Without it, calculations need more care: the equations $a\star x=b$ and $x\star a=b$ may have different solutions. This returns later with the product of matrices, which is generally not commutative.

Key Results¶

• An Abelian group is a group whose operation is commutative.

Examples¶

• Addition of real numbers is commutative. Matrix multiplication is generally not commutative.

Prerequisites¶

• Binary Operation

Related Concepts¶

• Group
• Field
• Matrix Product
• Distributivity

Sources¶

• Math II: Linear Algebra · Week 1

Aliases: commutative property, commutative law Symbols: element of

Last updated: 20 October 2026 03:02