Commutative Binary Operations

For binary operation

* : A × A → A

If (a, b) = (b, a)

Then it is commutative binary operation

Let's check some examples

Addition

+ : R × R → R

Since a + b = b + a

Hence, + is a commutative binary operation

Multiplication

× : R × R → R

Since a × b = b × a

Hence, × is a commutative binary operation

Subtraction

– : R × R → R

We have to check if

  a – b = b – a

Let a = 2 , b = 5

a – b = 2 – 5 = –3

b – a = 5 – 2 = 3

Since a – b ≠ b – a

Hence, – is not a commutative binary operation

Division

÷  : R * × R * → R *

Here R* is all real numbers except 0

We have to check if

  a ÷ b = b ÷ a

Let a = 2 , b = 5

a ÷ b = a/b = 2/5

b ÷ a = b/a = 5/2

Since a ÷ b ≠ b ÷ a

Hence, ÷ is not a commutative binary operation