# Abstract Algebra/Binary Operations

A binary operation on a set is a function . For , we usually write as . The property that for all is called *closure under *.

**Example:** Addition between two integers produces an integer result. Therefore addition is a binary operation on the integers. Whereas division of integers is an example of an operation that is *not* a binary operation. is not an integer, so the integers are not closed under division.

To indicate that a set has a binary operation defined on it, we can compactly write . Such a pair of a set and a binary operation on that set is collectively called a binary structure. A binary structure may have several interesting properties. The main ones we will be interested in are outlined below.

**Definition:** A binary operation on is *associative* if for all , .

**Example:** Addition of integers is associative: . Notice however, that subtraction is not associative. Indeed, .

**Definition:** A binary operation on is *commutative* is for all , .

**Example:** Multiplication of rational numbers is commutative: . Notice that division is not commutative: while . Notice also that commutativity of multiplication depends on the fact that multiplication of integers is commutative as well.

## Exercise[edit]

- Of the four arithmetic operations, addition, subtraction, multiplication, and division, which are associative? commutative and identity?

## Answer[edit]

operation | associative | commutative |
---|---|---|

Addition | yes | yes |

Multiplication | yes | yes |

Subtraction | No | No |

Division | No | No |