A binary operation on a set ${\displaystyle A}$ is a function ${\displaystyle *:A\times A\rightarrow A}$. For ${\displaystyle a,b\in A}$, we usually write ${\displaystyle *(a,b)}$ as ${\displaystyle a*b}$. The property that ${\displaystyle a*b\in A}$ for all ${\displaystyle a,b\in A}$ is called closure under ${\displaystyle *}$.

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. ${\displaystyle 1/2}$ is not an integer, so the integers are not closed under division.

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

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

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

Example: Multiplication of rational numbers is commutative: ${\displaystyle {\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {ac}{bd}}={\frac {ca}{bd}}={\frac {c}{d}}\cdot {\frac {a}{b}}}$. Notice that division is not commutative: ${\displaystyle 2\div 3={\frac {2}{3}}}$ while ${\displaystyle 3\div 2={\frac {3}{2}}}$. 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?

Answer