Abstract Algebra/Binary Operations

Definition

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?

Answer

operation	associative	commutative
Addition	yes	yes
Multiplication	yes	yes
Subtraction	No	No
Division	No	No

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