# Abstract Algebra/Binary Operations

A binary operation on a set $A$ is a function $*:A\times A\rightarrow A$ . For $a,b\in A$ , we usually write $*(a,b)$ as $a*b$ . The property that $a*b\in A$ for all $a,b\in A$ 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. $1/2$ is not an integer, so the integers are not closed under division.

To indicate that a set $A$ has a binary operation $*$ defined on it, we can compactly write $(A,*)$ . 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 $A$ is associative if for all $a,b,c\in A$ , $(a*b)*c=a*(b*c)$ .

Example: Addition of integers is associative: $(1+2)+3=6=1+(2+3)$ . Notice however, that subtraction is not associative. Indeed, $2=1-(2-3)\neq (1-2)-3=-4$ .

Definition: A binary operation $*$ on $A$ is commutative is for all $a,b\in A$ , $a*b=b*a$ .

Example: Multiplication of rational numbers is commutative: ${\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {ac}{bd}}={\frac {ca}{bd}}={\frac {c}{d}}\cdot {\frac {a}{b}}$ . Notice that division is not commutative: $2\div 3={\frac {2}{3}}$ while $3\div 2={\frac {3}{2}}$ . Notice also that commutativity of multiplication depends on the fact that multiplication of integers is commutative as well.

## Exercise

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