For binary operation

* : A × A → A

with identity element e

 

For element a in A,

there is an element b in A

such that

  a * b = e = b * a

Then, b is called inverse of a

 

Addition

+ : R × R R

 

For element a in A,

there is an element b in A

such that

  a * b = e = b * a

Then, b is called inverse of a

 

Here, e = 0 for addition

So, a * b = e = b * a

      a + b = 0 = b + a

⇒ b = –a

 

Since

  a + (– a) = 0 = (– a) + a,

So, –a is the inverse of a for addition.

 


Multiplication

× : R × R R

 

An element a in R is invertible if,

there is an element b in R such that ,

a * b = e = b * a

Here, b is the inverse of a

 

Here, e = 1 for multiplication

So, a * b = e = b * a

      a × b = 1 = b × a

⇒ b = 1/a

 

Since

  a × 1/a = 1 = 1/a × a

So,  1/a is the inverse of a for multiplication.

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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.