Identity element of Binary Operations

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 *