# NCERT Class 12-Mathematics: Chapter – 1 Relations and Functions Part 5 (For CBSE, ICSE, IAS, NET, NRA 2022)

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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**: