Class 12 Maths

Relation and Function

NCERT Exemplar Problem

Long Answer Type Part 3

Question 22: Each of the following defines a relation on N:

NCERT Exemplar Problems and Solution class 12 Math (46)

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

Solution:

NCERT Exemplar Problems and Solution class 12 Math (47)

Hence, the given relation is only transitive.


NCERT Exemplar Problems and Solution class 12 Math (48)

Therefore, R is not reflexive.

NCERT Exemplar Problems and Solution class 12 Math (49)

Therefore, R is symmetric.

NCERT Exemplar Problems and Solution class 12 Math (50)

Therefore, R is not transitive.

Thus, R is only symmetric.

NCERT Exemplar Problems and Solution class 12 Math (51)

Therefore, R is reflexive.

NCERT Exemplar Problems and Solution class 12 Math (52)

Therefore, it is clear that R is symmetric.

NCERT Exemplar Problems and Solution class 12 Math (53)

Therefore, R is transitive.

NCERT Exemplar Problems and Solution class 12 Math (54)

Therefore, R is not reflexive.

NCERT Exemplar Problems and Solution class 12 Math (55)

Therefore, R is not symmetric.

Since, there is no element which begins with NCERT Exemplar Problems and Solution class 12 Math (56)

Therefore, R is a transitive.

Question 23: Let A = {1, 2, 3, ………, 9} and R be the relation in A x A defined by (a, b) R (c, b) if a + d = b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation and also obtain the equivalent class [(2, 5)].

Solution:

NCERT Exemplar Problems and Solution class 12 Math (57)

Therefore, R is reflexive.



Let (a, b) R (c, d)

NCERT Exemplar Problems and Solution class 12 Math (58)

Therefore, R is symmetric.

Let (a, b) R (c, d) and (c, d) R (e, f)

NCERT Exemplar Problems and Solution class 12 Math (59)

Therefore, R is transitive.

Thus, R is reflexive, symmetric and transitive.

Therefore, R is an equivalence relation.

Equivalence class containing {(2, 5)} is {(1, 4), (2, 5), (3, 6), (4, 7), (5, 8), (6, 9)}.

Question 24: Using the definition, prove that the function NCERT Exemplar Problems and Solution class 12 Math (60) is invertible if and only if f is both one-one and onto.

Solution: By the definition of an invertible function:

A function NCERT Exemplar Problems and Solution class 12 Math (61) is defined to be and invertible function, if there exists a function NCERT Exemplar Problems and Solution class 12 Math (62)

The function g is called the inverse of f and is denoted by f – 1.

NCERT Exemplar Problems and Solution class 12 Math (63) has to one-one and onto.

Therefore, f(x) should be both one-one and onto.



Exercise 1

Exemplar Problems