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

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