Abstract Algebra/Binary Operations

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Definition[edit]

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[edit]

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

Answer[edit]

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