Binary operation

Definition

Let $S$ be a set. A binary operation on $S$ is a map $\ast \colon S \times S \to S$. For $a, b \in S$, the notation for the image of the pair $(a, b)$ is $a \ast b$.

Remarks

Addition, subtraction and multiplication are binary operations on the set $\mathbb{Z}$ of integers. Subtraction is not a binary operation on the set $\mathbb{N}$ of natural numbers, because $1 - 2$ is not in $\mathbb{N}$. The condition that $a \ast b$ is in $S$ for all $a, b \in S$ is often called closure.

Explanations

L1 human

A binary operation is a rule that takes two elements of a set, in order, and gives one element of the same set. For example, addition takes $2$ and $3$ and gives $5$. The order can matter: $2 - 3$ and $3 - 2$ are different.