**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) = 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}**

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

** x ^{2}**

** 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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