## HOW TO DETERMINE WHETHER A FUNCTION IS ONE TO ONE OR NOT

How to determine whether a function is one to one or not ?

Definition of one to one function :

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

Another name for one-to-one function is injective function

To prove a function f : A → B to be one-to-one, it is enough to prove any one of the following:

A function f : A → B is said to be one-to-one if x, y ∈ A, x  y ⇒ f(x)  f(y) [or equivalently f(x) = f(y) ⇒ x = y].

The following statements are some important simple results.

Let A and B be two sets with m and n elements.

(i) There is no one-to-one function from A to B if m > n.

(ii) If there is an one-to-one function from A to B, then m ≤ n.

Let us look into some example problems to understand the above concepts.

Example 1 :

Check whether the following function is one-to-one

f : N → N defined by f(n) = n + 2.

Solution :

To check if the given function is one to one, let us apply the rule

f (x)  =  f (y) ==> x = y

f (x)  =  x + 2 and f (y)   =  y + 2

So, x + 2  =  y + 2

x  =  y

For every element if set N has images in the set N. Hence it is one to one function.

Example 2 :

Check whether the following function is one-to-one

f : R → R defined by f(n) = n2

Solution :

To check if the given function is one to one, let us apply the rule

f (x)  =  f (y) ==> x = y

f (x)  =  x2 and f (y)   =  y2

By equating f (x) and f (y), we get

x2 =   y2

x  =  y

From this we cannot decide that the function is one to one. Because every two different elements in the domain has same images is co-domain. That is,

If x = 1 then y = 1

If x = -1 then y is also 1.

Hence the given function is not one to one.

Example 3 :

Check whether the following function is one-to-one

f : R - {0} → R defined by f(x) = 1/x

Solution :

To check if the given function is one to one, let us apply the rule

f (x)  =  f (y) ==> x = y

f (x)  =  1/x and f (y)  =  1/y

By applying the above rule, we get

1/x  =  1/y

x  =  y

Every element in domain has different images in co-domain. Hence it is one to one function.

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