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