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

(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 ________

.

Question 26:

The domain of the function defined by is ________.

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 ________.

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.

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 ________.

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.

Question 31:

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

Question 32:

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

Question 33:

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

Question 34:

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

1.3 EXERCISE

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.

Question 2:

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