# Relations and Functions-II, Inverse of a Function, Binary Operations

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## Inverse of a Function

Consider the relation

This is a many-to-one function. Now let us find the inverse of this relation. Pictorially, it can be represented as

Clearly this relation does not represent a function. (Why?)

Now take another relation

It represents one-to-one onto function. Now let us find the inverse of this relation, which is represented pictorially as

This represents a function.

Consider the relation

If represents many-to-one function. Now find the inverse of the relation. Pictorially it is represented as

This does not represent a function, because element of set is not associated with any element of . Also note that the elements of does not have a unique image.

Let us take the following relation

It represent one-to-one into function. Find the inverse of the relation.

It does not represent a function because the element of is not associated with any element of . From the above relations we see that we may or may not get a relation as a function when we find the inverse of a relation (function).

We see that the inverse of a function exists only if the function is one-to-one onto function i.e. only if it is a bijective function.

## Binary Operations:

Let be two non-empty sets, then a function from to is called a binary operation on .

If a binary operation on is denoted by , the unique element of associated with the ordered pair of is denoted by .

The order of the elements is taken into consideration, i.e. the elements associated with the pairs and may be different i.e. may not be equal to .

Let be a non-empty set and be an operation on , then

is said to be closed under the operation if for all implies .

The operation is said to be commutative if for all .

The operation is said to be associative if for all .

An element is said to be an identity element iff

An element is called invertible iff these exists some such that is called inverse of .