Class 12 Mathematics

Relation and Function

NCERT Exemplar Problems and Solution

NCERT Exemplar Problems-Long Answer-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.


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.


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