# NCERT Class 12-Mathematics: Chapter –1 Relations and Functions Part 10

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**Question 21:**

Let . Then, discuss whether the following functions defined on A are one-one, onto or bijective:

(i)

(ii)

(iii)

(iv)

**Answer:**

(i) is one-one but not onto

(ii) is neither one-one nor onto

(iii) is bijective,

(iv) is neither one-one nor onto.

**Question 22:**

Each of the following defines a relation on **N**:

(i) is greater than

(ii)

(iii) is square of an integer

(iv) .

Determine which of the above relations are reflexive, symmetric and transitive.

**Answer:**

(i) Transitive

(ii) Symmetric

(iii) Reflexive, symmetric and transitive

(iv) Transitive.

**Question 23:**

Let and R be the relation in defined by for in . Prove that R is an equivalence relation and also obtain the equivalent class .

**Answer:**

**Question 24:**

Using the definition, prove that the function is invertible if and only if is both one-one and onto.

**Answer:**

is one-one if the images of distinct elements of A under f are distinct, i.e., for every ,

We suppose that is not one-one function.

Let and

and

Inverse function cannot be defined as we have two images ‘a’ and ‘b’ for one pre-image ‘x’.

So, f can be invertible if it is one-one.

Now, suppose that is not onto function.

Let and range of

We can observe that ‘z’ does not have any pre-image in A.

But has z as a pre-image which does not have any image in A.

So, f can be invertible if it is onto.

Hence, f is invertible if and only if it is both one-one and onto.

**Question 25:**

Functions are defined, respectively, by, find

(i)

(ii)

(iii)

(iv)

**Answer:**

(i)

(ii)

(iii)

(iv)