INJECTIVE FUNCTION

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

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

That is, we say f is one to 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.

Solved Problems

Problem 1 : 

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)}

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

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 A has a different image in B.

That is, no two or more elements of A have the same image in B.

Therefore, f is one to one or injective function.

Problem 2 : 

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)}

Is f injective ? Explain.  

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. 

The elements "a" and "c" in X have the same image "e" in Y. 

So, f is not one to one or injective. 

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 injective ? 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 A has a different image in B.

That is, no two or more elements of A have the same image in B.

Therefore, f is one to one or injective function.

Moreover, the above mapping is one to one and onto or bijective function.  

Related Topics

Onto or Surjective function

One to one and Onto or Bijective function

Into function

Constant Function

Identity function

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