# NCERT Class 12-Mathematics: Chapter –1 Relations and Functions Part 5

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**Question 24:**

For real numbers *x* and , define if and only if is an irrational number. Then the relation R is

(A) Reflexive

(B) Symmetric

(C) Transitive

(D) None of these

**Answer:**

(A) Is the correct choice.

## Fill in the Blanks in Each of the Examples 25 to 30

**Question 25:**

Consider the set and R be the smallest equivalence relation on A, then ________

**Answer:**

.

**Question 26:**

The domain of the function defined by is ______________.

**Answer:**

Here

Hence the domain of

**Question 27:**

Consider the set A containing elements. Then, the total number of injective functions from A onto itself is ________.

**Answer:**

**Question 28:**

Let **Z** be the set of integers and R be the relation defined in **Z** such that if is divisible by . Then R partitions the set **Z** into ________ pairwise disjoint subsets.

**Answer:**

Three

**Question 29:**

Let **R** be the set of real numbers and be the binary operation defined on **R** as . Then, the identity element with respect to the binary operation is _______.

**Answer:**

is the identity element with respect to the binary operation .

State **True** or **False** for the statements in each of the Examples 30 to 34.

**Question 30:**

Consider the set and the relation . R is a transitive relation.

**Answer:** True.

**Question 31:**

Let A be a finite set. Then, each injective function from A into itself is not surjective.

**Answer:** False

**Question 32:**

For sets A, B and C, let be functions such that is injective. Then both and are injective functions.

**Answer:** False

**Question 33:**

For sets A, B and C, let be functions such that is surjective. Then is surjective

**Answer:** True

**Question 34:**

Let **N** be the set of natural numbers. Then, the binary operation in **N** defined as has identity element.

**Answer:** False

## 1.3 EXERCISE

### Short Answer (S.A)

**Question 1:**

Let and the relation R be defined on A as follows:

.

Then, write minimum number of ordered pairs to be added in **R** to make **R** reflexive and transitive.

**Answer:**

**Question 2:**

Let D be the domain of the real valued function *f* defined by . Then, write D.

**Answer:**

**Question 3:**

Let be defined by and, respectively. Then, find .

**Answer:**