**Injective surjective and bijective functions :**

Here, we are going to see the different types of functions on sets.

(i) One to one or Injective function

(ii) Onto or Surjective function

(iii) One to one and onto or Bijective function

Let us discuss the above different types of functions in detail.

Let f : A ----> B be a function.

The function f is called an one-one function, if it takes different elements of A into different elements of B.

That is, we say f is one-one

In other words f is one-one if no element in B is associated with more than one element in A.

A one-one function is also called an Injective function.

The figure given below represents a one-one function.

Let f : A ----> B be a function.

The function f is called an onto function, if every element in B has a pre-image in A.

That is, in B all the elements will be involved in mapping.

An onto function is also called a surjective function.

The figure given below represents a onto function.

Let f : A ----> B be a function.

The function f is called an one-one and onto or a bijective function if f is both a one-one and an onto function

More clearly,

f maps distinct elements of A into distinct images in B and every element in B is an image of some element in A.

The figure given below represents a one to one and onto or bijective function.

Apart from "Injective surjective and bijective functions", we have some other functions on sets.

**Let us discuss the other different types of functions here :**

The other different types of functions on sets are

(i) Constant function

(ii) Identity function

Let us discuss the above different types of functions in detail.

The function f is called constant function if every element of A has the same image in B.

Range of a constant function is a singleton set.

Let A = { x, y, u, v, 1 }, B = { 3, 5, 7, 8, 10, 15 }.

The function f : A ---> B defined by f (x) = 5 for every x belonging to A is a constant function.

The figure given below represents a constant function.

Let A be a non-empty set. A function f : A ---> A is called an identity function of A if f (a) = a for all a belonging to A.

That is, an identity function maps each element of A into itself.

For example, let A be the set of real numbers (R). The function f : R ----> R be defined by f (x) = x for all x belonging to R is the identity function on R.

The figure given below represents the graph of the identity function on R.

Let f : A ----> B be a function.

Then, we have

**Domain : Set A**

**Co-domain : Set B**

**Range : Elements of B involved in mapping. **

Note :

In onto function, co-domain = Range

After having gone through the stuff given above, we hope that the students would have understood "Injective surjective and bijective functions".

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