For binary operation

* : A × A → A

 

e is called identity of * if

  a * e  = e * a = a

Here e is called identity element of binary operation.

 


Addition

+ : R × R R

 

e is called identity of * if

  a * e  = e * a = a

i.e.

  a + e = e + a = a

 

This is only possible if e = 0

 

Since a + 0 = 0 + a = a ∀ a ∈ R

0 is the identity element for addition on R

 


Multiplication

e is the identity of * if

a * e  = e * a = a

i.e. a × e = e × a = a

This is possible if e = 1

 

Since a × 1 = 1 × a = a ∀ a ∈ R

1 is the identity element for multiplication on R

 


Subtraction

e is the identity of * if

a * e  = e * a = a

i.e. a – e = e – a = a

There is no possible value of e where a – e = e – a

 

So, subtraction has no identity element in R

 


Division

e is the identity of * if

a * e  = e * a = a

i.e. a/e = e/a = a

There is no possible value of e where a/e = e/a = a

 

So, division has no identity element in R *

  1. Chapter 1 Class 12 Relation and Functions
  2. Concept wise
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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.