Whether binary commutative/associative or not

Chapter 1 Class 12 Relation and Functions
Concept wise

For binary operation

* : A × A → A

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

Then it is commutative binary operation

Let's check some examples

+ : 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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#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.