Binary Operation¶

Definition¶

A binary operation $\star$ on a set $A$ is a function

\star\colon A\times A\to A.

Interpretation¶

A binary operation combines two elements of a set into an element of the same set. Writing it as a function $\star\colon A\times A\to A$ makes this precise.

The domain $A\times A$ is the Cartesian product: the input is an ordered pair $(a,b)$ , and $a\star b$ is short for $\star(a,b)$ . Because the pair is ordered, $a\star b$ and $b\star a$ can differ. Because $\star$ is a function, every pair has exactly one output. The codomain $A$ expresses closure: the result never leaves the set.

Whether a rule is a binary operation therefore depends on the set. Subtraction is a binary operation on $\mathbb{Z}$ but not on $\mathbb{N}$ , and division is one on the nonzero rational numbers but not on $\mathbb{Q}$ , where $a/0$ has no value.

Key Results¶

Closure is built into the definition: $a\star b\in A$ for all $a,b\in A$ $A$ .
Associativity, identity elements, and inverse elements determine which algebraic structure is present.

Examples¶

Addition on $\mathbb{R}$ is a binary operation. Subtraction on $\mathbb{N}$ is not a binary operation on $\mathbb{N}$ because the result may be negative.

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 1

Aliases: binary operator, dyadic operation Symbols: element of, integers, natural numbers, rational numbers, real numbers

Last updated: 20 October 2026 03:02