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
**
**
*
**

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