SURJECTIVE 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.

Solved Problems

Problem 1 : 

Let f : X ----> Y. X, Y and f are defined as 

X  =  {a, b, c, d}

Y  =  {d, e, f}

f  =  {(a, e), (b, f), (c, e), (d, d)}

Verify whether f is a function. if so, what type of function is f ?

In the above arrow diagram, all the elements of X have images in Y and every element of X has a unique image.

That is, no element of X has more than one image.

So, f is a function. 

Every element of Y has a pre-image in X. 

Therefore, f is onto or surjective function. 

Problem 2 : 

Let f : A ----> B. A, B and f are defined as 

A  =  {1, 2, 3}

B  =  {5, 6, 7, 8}

f  =  {(1, 5), (2, 8), (3, 6)}

Is f surjective ? Explain. 

Solution :

Write the elements of f (ordered pairs) using arrow diagram as shown below

In the above arrow diagram, all the elements of A have images in B and every element of A has a unique image.

That is, no element of A has more than one image.

So, f is a function. 

The element "7" in B has no pre-image in A. 

Because the element "7" has no pre-image, f is not onto or surjective function. 

Problem 3 : 

Let f : A ----> B. A, B and f are defined as 

A  =  {1, 2, 3, 4}

B  =  {5, 6, 7, 8}

f  =  {(1, 8), (2, 6), (3, 5), (4, 7)}

Is f surjective ? Explain. 

Solution :

Write the elements of f (ordered pairs) using arrow diagram as shown below. 

In the above arrow diagram, all the elements of A have images in B and every element of A has a unique image.

That is, no element of A has more than one image.

So, f is a function. 

Every element of B has a pre- image in A.

Therefore, f is onto or surjective function.

Related Topics

One to one or Injective function

One to one and Onto or Bijective function

Into function

Constant Function

Identity function

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