# 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

Constant Function

Identity function

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