Inverse of Binary Operations

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.