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**Formal Definition**

A **binary operation** $*$ on a set $S$ is a function mapping $S\times S$ into $S$. For each $(a,b)\in S\times S$, we will denote the element $*((a,b))$ of $S$ by $a*b$.

**Informal Definition**

A binary operation on a set is a calculation performed on two elements of the set to produce another element of the set.

**Example(s)**

Our usual addition $+$ is a binary operation on the set $\mathbb{R}$. Our usual multiplication $\cdot$ is a different binary operation on $\mathbb{R}$. In this example, we could replace $\mathbb{R}$ by any of the sets $\mathbb{C}, \mathbb{Z}, \mathbb{R^+}, or\ \mathbb{Z^+}$.

**Non-example(s)**

Let $M(\mathbb{R})$ be the set of all matrices with real entries.The usual matrix addition $+$ is not a binary operation on this set since $A + B$ is not defined for an ordered pair $(A, B)$ of matrices having different numbers of rows or of columns.

**Additional Comments**

Sometimes a binary operation on $S$ provides a binary operation on a subset $H$ of $S$ also.