Abstract Algebra/Binary Operations

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Definition[edit]

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