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

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Inverse of a Function

Consider the relation

Consider the relation

Consider the Relation

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

Consider the relation does not represent

Consider the Relation Does Not Represent

Consider the relation does not represent

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

Now take another relation

One-to-one onto function

One-to-One Onto Function

One-to-one onto function

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

One-to-one onto inverse function

One-to-One Onto Inverse Function

One-to-one onto inverse function

This represents a function.

Consider the relation

Many-to-one function

Many-to-One Function

Many-to-one function

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

Many-to-one inverse function

Many-to-One Inverse Function

Many-to-one inverse function

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

One-to-one into function

One-to-One into Function

One-to-one into function

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

One-to-one into inverse function

One-to-One into Inverse Function

One-to-one into inverse function

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 .

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