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# RD Sharma solutions for Class 12 Mathematics chapter 1 - Relations

## Chapter 1: Relations

Ex. 1.1Ex. 1.2Ex. 1.3Ex. 1.4Ex. 1.5

#### Chapter 1: Relations Exercise 1.1 solutions [Pages 10 - 11]

Ex. 1.1 | Q 1.1 | Page 10

Let A be the set of all human beings in a town at a particular time. Determine whether of the following relations are reflexive, symmetric and transitive :

= {(xy) : x and y work at the same place}

Ex. 1.1 | Q 1.2 | Page 10

Let A be the set of all human beings in a town at a particular time. Determine whether of the following relations are reflexive, symmetric and transitive :

R = {(xy) : x and y live in the same locality}

Ex. 1.1 | Q 1.3 | Page 10

Let A be the set of all human beings in a town at a particular time. Determine whether of the following relations are reflexive, symmetric and transitive:

R = {(xy) : x is wife of y}

Ex. 1.1 | Q 1.4 | Page 10

Let A be the set of all human beings in a town at a particular time. Determine whether of the following relations are reflexive, symmetric and transitive:

R = {(xy) : is father of and y}

Ex. 1.1 | Q 2 | Page 10

Three relations R1, R2 and R3 are defined on a set A = {a, b, c} as follows:
R1 = {(a, a), (a, b), (a, c), (b, b), (b, c), (c, a), (c, b), (c, c)}
R2 = {(a, a)}
R3 = {(b, c)}
R4 = {(a, b), (b, c), (c, a)}.

Find whether or not each of the relations R1, R2R3R4 on is (i) reflexive (ii) symmetric and (iii) transitive.

Ex. 1.1 | Q 3.1 | Page 10

Test whether the following relations R1 are  (i) reflexive (ii) symmetric and (iii) transitive :

R1 on Q0 defined by (a, b) ∈ R1 ⇔ = 1/b.

Ex. 1.1 | Q 3.2 | Page 10

Test whether the following relations R2 are (i) reflexive (ii) symmetric and (iii) transitive:

R2 on Z defined by (a, b) ∈ R2 ⇔ |a – b| ≤ 5

Ex. 1.1 | Q 3.3 | Page 10

Test whether the following relations R3 are (i) reflexive (ii) symmetric and (iii) transitive:

R3 on R defined by (a, b) ∈ R3 ⇔ a2 – 4ab + 3b2 = 0.

Ex. 1.1 | Q 4 | Page 10

Let A = {1, 2, 3}, and let R1 = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}, R2 = {(2, 2), (3, 1), (1, 3)}, R3 = {(1, 3), (3, 3)}. Find whether or not each of the relations R1, R2, R3 on A is (i) reflexive (ii) symmetric (iii) transitive.

Ex. 1.1 | Q 5.1 | Page 11

The following relations are defined on the set of real numbers.
aRb if a – b > 0

Find whether relations are reflexive, symmetric or transitive.

Ex. 1.1 | Q 5.2 | Page 11

The following relations are defined on the set of real numbers.

aRb if 1 + ab > 0

Find whether relations are reflexive, symmetric or transitive.

Ex. 1.1 | Q 5.3 | Page 11

The following relations is defined on the set of real numbers.  aRb if |a| ≤ b

Find whether relations are reflexive, symmetric or transitive.

Ex. 1.1 | Q 6 | Page 11

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(ab): b = a + 1} is reflexive, symmetric or transitive.

Ex. 1.1 | Q 7 | Page 11

Check whether the relation R in R defined as R = {(ab): a ≤ b3} is reflexive, symmetric or transitive.

Ex. 1.1 | Q 8 | Page 11

Prove that every identity relation on a set is reflexive, but the converse is not necessarily true.

Ex. 1.1 | Q 9.1 | Page 11

If = {1, 2, 3, 4} define relations on A which have properties of being reflexive, transitive but not symmetric ?

Ex. 1.1 | Q 9.2 | Page 11

If = {1, 2, 3, 4} define relations on A which have properties of being symmetric but neither reflexive nor transitive ?

Ex. 1.1 | Q 9.3 | Page 11

If = {1, 2, 3, 4} define relations on A which have properties of being reflexive, symmetric and transitive. ?

Ex. 1.1 | Q 10 | Page 11

Let R be a relation defined on the set of natural numbers N as
R = {(xy) : x N, 2x + y = 41}
Find the domain and range of R. Also, verify whether R is (i) reflexive, (ii) symmetric (iii) transitive.

Ex. 1.1 | Q 11 | Page 11

Is it true that every relation which is symmetric and transitive is also reflexive? Give reasons.

Ex. 1.1 | Q 12 | Page 11

An integer m is said to be related to another integer n if m is a multiple of n.Check if the relation is symmetric, reflexive and transitive.

Ex. 1.1 | Q 13 | Page 11

Show that the relation '≥' on the set R of all real numbers is reflexive and transitive but not symmetric ?

Ex. 1.1 | Q 14.1 | Page 11

Give an example of a relation which is reflexive and symmetric but not transitive ?

Ex. 1.1 | Q 14.2 | Page 11

Give an example of a relation which is reflexive and transitive but not symmetric ?

Ex. 1.1 | Q 14.3 | Page 11

Give an example of a relation which is symmetric and transitive but not reflexive ?

Ex. 1.1 | Q 14.4 | Page 11

Give an example of a relation which is symmetric but neither reflexive nor transitive ?

Ex. 1.1 | Q 14.5 | Page 11

Give an example of a relation which is transitive but neither reflexive nor symmetric ?

Ex. 1.1 | Q 15 | Page 11

Given the relation R = {(1, 2), (2, 3)} on the set A = {1, 2, 3}, add a minimum number of ordered pairs so that the enlarged relation is symmeteric, transitive and reflexive.

Ex. 1.1 | Q 16 | Page 11

Let A = {1, 2, 3} and R = {(1, 2), (1, 1), (2, 3)} be a relation on A. What minimum number of ordered pairs may be added to R so that it may become a transitive relation on A.

Ex. 1.1 | Q 17 | Page 11

Let A = {abc} and the relation R be defined on A as follows: R = {(aa), (bc), (ab)}. Then, write minimum number of ordered pairs to be added in R to make it reflexive and transitive.                                                                                                                                        [NCERT EXEMPLAR]

Ex. 1.1 | Q 18.1 | Page 11

Defines a relation on :
x > yxy ∈  N

Determine which of the above relations are reflexive, symmetric and transitive.                                                                                                                      [NCERT EXEMPLAR]

Ex. 1.1 | Q 18.2 | Page 11

Defines a relation on :

x + y = 10, xy∈ N

Determine which of the above relations are reflexive, symmetric and transitive.                                                                                                                                 [NCERT EXEMPLAR]

Ex. 1.1 | Q 18.3 | Page 11

Defines a relation on :

xy is square of an integer, xy ∈ N

Determine which of the above relations are reflexive, symmetric and transitive.                                                                                                         [NCERT EXEMPLAR]

Ex. 1.1 | Q 18.4 | Page 11

Defines a relation on N:

x + 4y = 10, xy ∈ N

Determine which of the above relations are reflexive, symmetric and transitive.                                                                                 [NCERT EXEMPLAR]

#### Chapter 1: Relations Exercise 1.2 solutions [Pages 26 - 27]

Ex. 1.2 | Q 1 | Page 26

Show that the relation R defined by R = {(a, b) : a – b is divisible by 3; a, b ∈ Z} is an equivalence relation.

Ex. 1.2 | Q 2 | Page 26

Show that the relation R on the set Z of integers, given by
R = {(a, b) : 2 divides a – b},  is an equivalence relation.

Ex. 1.2 | Q 3 | Page 26

Prove that the relation R on Z defined by
(ab) ∈ R ⇔ a − b is divisible by 5
is an equivalence relation on Z.

Ex. 1.2 | Q 4 | Page 26

Let n be a fixed positive integer. Define a relation R on Z as follows:
(ab) ∈ R ⇔ a − b is divisible by n.
Show that R is an equivalence relation on Z.

Ex. 1.2 | Q 5 | Page 26

Let Z be the set of integers. Show that the relation
R = {(ab) : ab ∈ Z and a + b is even}
is an equivalence relation on Z.

Ex. 1.2 | Q 6 | Page 26

m is said to be related to n if m and n are integers and m − n is divisible by 13. Does this define an equivalence relation?

Ex. 1.2 | Q 7 | Page 26

Let be a relation on the set A of ordered pair of integers defined by (xyR (uv) if xv = yu. Show that R is an equivalence relation.

Ex. 1.2 | Q 8 | Page 26

Show that the relation R on the set A = {∈ Z ; 0 ≤ x ≤ 12}, given by R = {(ab) : a = b}, is an equivalence relation. Find the set of all elements related to 1.

Ex. 1.2 | Q 9 | Page 27

Let L be the set of all lines in XY-plane and R be the relation in L defined as R = {L1L2) : L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y= 2x + 4.

Ex. 1.2 | Q 10 | Page 27

Show that the relation R, defined on the set A of all polygons as
R = {(P1P2) : P1 and P2 have same number of sides},
is an equivalence relation. What is the set of all elements in A related to the right angle triangle Twith sides 3, 4 and 5?

Ex. 1.2 | Q 11 | Page 27

Let O be the origin. We define a relation between two points P and Q in a plane if OP = OQ. Show that the relation, so defined is an equivalence relation.

Ex. 1.2 | Q 12 | Page 27

Let R be the relation defined on the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(ab) : both a and b are either odd or even}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each other and all the elements of the subset {2, 4, 6} are related to each other, but no element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}.

Ex. 1.2 | Q 13 | Page 27

Let S be a relation on the set R of all real numbers defined by
S = {(ab) ∈ R × R : a2 + b2 = 1}
Prove that S is not an equivalence relation on R.

Ex. 1.2 | Q 14 | Page 27

Let Z be the set of all integers and Z0 be the set of all non-zero integers. Let a relation R on Z × Z0be defined as (ab(cd) ⇔ ad = bc for all (ab), (cd) ∈ Z × Z0,
Prove that R is an equivalence relation on Z × Z0.

Ex. 1.2 | Q 15.1 | Page 27

If R and S are relations on a set A, then prove that R and S are symmetric ⇒ R ∩ S and R ∪ S are symmetric ?

Ex. 1.2 | Q 15.2 | Page 27

If R and S are relations on a set A, then prove that R is reflexive and S is any relation ⇒ R ∪ S is reflexive ?

Ex. 1.2 | Q 16 | Page 27

If R and are transitive relations on a set A, then prove that R ∪ S may not be a transitive relation on A.

Ex. 1.2 | Q 17 | Page 27

Let C be the set of all complex numbers and Cbe the set of all no-zero complex numbers. Let a relation R on Cbe defined as

z_1 R  z_2  ⇔ (z_1 -z_2)/(z_1 + z_2)  is real for all z1, z2 ∈C0 .

Show that R is an equivalence relation.

#### Chapter 1: Relations Exercise 1.3 solutions [Pages 29 - 30]

Ex. 1.3 | Q 1 | Page 29

Write the domain of the relation R defined on the set Z of integers as follows:
(ab) ∈ R ⇔ a2 + b2 = 25

Ex. 1.3 | Q 2 | Page 30

If R = {(xy) : x2 + y2 ≤ 4; xy ∈ Z} is a relation on Z, write the domain of R.

Ex. 1.3 | Q 3 | Page 30

Write the identity relation on set A = {abc}.

Ex. 1.3 | Q 4 | Page 30

Write the smallest reflexive relation on set A = {1, 2, 3, 4}.

Ex. 1.3 | Q 5 | Page 30

If R = {(xy) : x + 2y = 8} is a relation on N by, then write the range of R.

Ex. 1.3 | Q 6 | Page 30

If R is a symmetric relation on a set A, then write a relation between R and R−1.

Ex. 1.3 | Q 7 | Page 30

Let R = {(xy) : |x2 − y2| <1) be a relation on set A = {1, 2, 3, 4, 5}. Write R as a set of ordered pairs.

Ex. 1.3 | Q 8 | Page 30

If A = {2, 3, 4}, B = {1, 3, 7} and R = {(xy) : x ∈ Ay ∈ B and x < y} is a relation from A to B, then write R−1.

Ex. 1.3 | Q 9 | Page 30

Let A = {3, 5, 7}, B = {2, 6, 10} and R be a relation from A to B defined by R = {(xy) : x and y are relatively prime}. Then, write R and R−1.

Ex. 1.3 | Q 10 | Page 30

Define a reflexive relation ?

Ex. 1.3 | Q 11 | Page 30

Define a symmetric relation ?

Ex. 1.3 | Q 12 | Page 30

Define a transitive relation ?

Ex. 1.3 | Q 13 | Page 30

Define an equivalence relation ?

Ex. 1.3 | Q 14 | Page 30

If A = {3, 5, 7} and B = {2, 4, 9} and R is a relation given by "is less than", write R as a set ordered pairs.

Ex. 1.3 | Q 15 | Page 30

A = {1, 2, 3, 4, 5, 6, 7, 8} and if R = {(xy) : y is one half of xxy ∈ A} is a relation on A, then write R as a set of ordered pairs.

Ex. 1.3 | Q 16 | Page 30

Let A = {2, 3, 4, 5} and B = {1, 3, 4}. If R is the relation from A to B given by a R b if "a is a divisor of b". Write R as a set of ordered pairs.

Ex. 1.3 | Q 17 | Page 30

State the reason for the relation R on the set {1, 2, 3} given by R = {(1, 2), (2, 1)} to be transitive ?

Ex. 1.3 | Q 18 | Page 30

Let R = {(aa3) : a is a prime number less than 5} be a relation. Find the range of R.

[CBSE 2014]

Ex. 1.3 | Q 19 | Page 30

Let R be the equivalence relation on the set Z of the integers given by R = { (ab) : 2 divides }.

Write the equivalence class [0].

Ex. 1.3 | Q 20 | Page 30

For the set A = {1, 2, 3}, define a relation R on the set A as follows:
R = {(1, 1), (2, 2), (3, 3), (1, 3)}
Write the ordered pairs to be added to R to make the smallest equivalence relation.

Ex. 1.3 | Q 21 | Page 30

Let A = {0, 1, 2, 3} and R be a relation on A defined as
R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}
Is R reflexive? symmetric? transitive?

Ex. 1.3 | Q 22 | Page 30

Let the relation R be defined on the set A = {1, 2, 3, 4, 5} by R = {(ab) : | a2b| < 8}. Write as a set of ordered pairs.

Ex. 1.3 | Q 23 | Page 30

Let the relation R be defined on N by aRb iff 2a + 3b = 30. Then write R as a set of ordered pairs

Ex. 1.3 | Q 24 | Page 30

Write the smallest equivalence relation on the set A = {1, 2, 3} ?

#### Chapter 1: Relations Exercise 1.4 solutions [Pages 31 - 32]

Ex. 1.4 | Q 1 | Page 31

Let R be a relation on the set N given by
R = {(ab) : a = b − 2, b > 6}. Then,
(a) (2, 4) ∈ R
(b) (3, 8) ∈ R
(c) (6, 8) ∈ R
(d) (8, 7) ∈ R

Ex. 1.4 | Q 2 | Page 31

If a relation R is defined on the set Z of integers as follows:
(ab) ∈ R ⇔ a2 + b2 = 25. Then, domain (R) is
(a) {3, 4, 5}
(b) {0, 3, 4, 5}
(c) {0, ± 3, ± 4, ± 5}
(d) none of these

Ex. 1.4 | Q 3 | Page 31

R is a relation on the set Z of integers and it is given by
(xy) ∈ R ⇔ | x − y | ≤ 1. Then, R is
(a) reflexive and transitive
(b) reflexive and symmetric
(c) symmetric and transitive
(d) an equivalence relation

Ex. 1.4 | Q 4 | Page 31

The relation R defined on the set A = {1, 2, 3, 4, 5} by
R = {(ab) : | a2 − b2 | < 16} is given by
(a) {(1, 1), (2, 1), (3, 1), (4, 1), (2, 3)}
(b) {(2, 2), (3, 2), (4, 2), (2, 4)}
(c) {(3, 3), (4, 3), (5, 4), (3, 4)}
(d) none of these

Ex. 1.4 | Q 5 | Page 31

Let R be the relation over the set of all straight lines in a plane such that  l1 l2 ⇔ 1⊥ l2. Then, Ris
(a) symmetric
(b) reflexive
(c) transitive
(d) an equivalence relation

Ex. 1.4 | Q 6 | Page 31

If A = {abc}, then the relation R = {(bc)} on A is
(a) reflexive only
(b) symmetric only
(c) transitive only
(d) reflexive and transitive only

Ex. 1.4 | Q 7 | Page 31

Let A = {2, 3, 4, 5, ..., 17, 18}. Let 'â‰ƒ' be the equivalence relation on A × A, cartesian product of Awith itself, defined by (ab) â‰ƒ (cd) if ad = bc. Then, the number of ordered pairs of the equivalence class of (3, 2) is
(a) 4
(b) 5
(c) 6
(d) 7

Ex. 1.4 | Q 8 | Page 31

Let A = {1, 2, 3}. Then, the number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is
(a) 1
(b) 2
(c) 3
(d) 4

Ex. 1.4 | Q 9 | Page 31

The relation 'R' in N × N such that
(abR (cd) ⇔ a + d = b + c is
(a) reflexive but not symmetric
(b) reflexive and transitive but not symmetric
(c) an equivalence relation
(d) none of the these

Ex. 1.4 | Q 10 | Page 32

If A = {1, 2, 3}, B = {1, 4, 6, 9} and R is a relation from A to B defined by 'x is greater than y'. The range of R is
(a) {1, 4, 6, 9}
(b) {4, 6, 9}
(c) {1}
(d) none of these

Ex. 1.4 | Q 11 | Page 32

A relation R is defined from {2, 3, 4, 5} to {3, 6, 7, 10} by : x R y ⇔ x is relatively prime to y. Then, domain of R is
(a) {2, 3, 5}
(b) {3, 5}
(c) {2, 3, 4}
(d) {2, 3, 4, 5}

Ex. 1.4 | Q 12 | Page 32

A relation Ï• from C to R is defined by x Ï• y ⇔ | x | = y. Which one is correct?
(a) (2 + 3 i) Ï• 13
(b) 3 Ï• (−3)
(c) (1 + i) Ï• 2
(d) i Ï• 1

Ex. 1.4 | Q 13 | Page 32

Let R be a relation on N defined by + 2y = 8. The domain of R is
(a) {2, 4, 8}
(b) {2, 4, 6, 8}
(c) {2, 4, 6}
(d) {1, 2, 3, 4}

Ex. 1.4 | Q 14 | Page 32

R is a relation from {11, 12, 13} to {8, 10, 12} defined by y = x − 3. Then, R−1 is
(a) {(8, 11), (10, 13)}
(b) {(11, 8), (13, 10)}
(c) {(10, 13), (8, 11)}
(d) none of these

Ex. 1.4 | Q 15 | Page 32

Let R = {(aa), (bb), (cc), (ab)} be a relation on set A = abc. Then, R is
(a) identify relation
(b) reflexive
(c) symmetric
(d) antisymmetric

Ex. 1.4 | Q 16 | Page 32

Let A = {1, 2, 3} and B = {(1, 2), (2, 3), (1, 3)} be a relation on A. Then, R is
(a) neither reflexive nor transitive
(b) neither symmetric nor transitive
(c) transitive
(d) none of these

Ex. 1.4 | Q 17 | Page 32

If R is the largest equivalence relation on a set A and S is any relation on A, then
(a) R ⊂ S
(b) S ⊂ R
(c) R = S
(d) none of these

Ex. 1.4 | Q 18 | Page 32

If R is a relation on the set A = {1, 2, 3, 4, 5, 6, 7, 8, 9} given by x R y ⇔ y = 3 x, then R =
(a) {(3, 1), (6, 2), (8, 2), (9, 3)}
(b) {(3, 1), (6, 2), (9, 3)}
(c) {(3, 1), (2, 6), (3, 9)}
(d) none of these

Ex. 1.4 | Q 19 | Page 32

If R is a relation on the set A = {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3)}, then R is
(a) reflexive
(b) symmetric
(c) transitive
(d) all the three options

Ex. 1.4 | Q 20 | Page 32

If A = {abcd}, then a relation R = {(ab), (ba), (aa)} on A is
(a) symmetric and transitive only
(b) reflexive and transitive only
(c) symmetric only
(d) transitive only

Ex. 1.4 | Q 21 | Page 32

If A = {1, 2, 3}, then a relation R = {(2, 3)} on A is
(a) symmetric and transitive only
(b) symmetric only
(c) transitive only
(d) none of these

Ex. 1.4 | Q 22 | Page 32

Let R be the relation on the set A = {1, 2, 3, 4} given by
R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Then,
(a) R is reflexive and symmetric but not transitive
(b) R is reflexive and transitive but not symmetric
(c) is symmetric and transitive but not reflexive
(d) R is an equivalence relation

Ex. 1.4 | Q 23 | Page 32

Let A = {1, 2, 3}. Then, the number of equivalence relations containing (1, 2) is
(a) 1
(b) 2
(c) 3
(d) 4

Ex. 1.4 | Q 24 | Page 32

The relation R = {(1, 1), (2, 2), (3, 3)} on the set {1, 2, 3} is
(a) symmetric only
(b) reflexive only
(c) an equivalence relation
(d) transitive only

#### Chapter 1: Relations Exercise 1.5 solutions [Pages 31 - 32]

Ex. 1.5 | Q 1.1 | Page 31

Give an example of a function which is one-one but not onto ?

Ex. 1.5 | Q 1.2 | Page 31

Give an example of a function which is not one-one but onto ?

Ex. 1.5 | Q 1.3 | Page 31

Give an example of a function which is neither one-one nor onto ?

Ex. 1.5 | Q 2.1 | Page 31

Which of the following functions from A to B are one-one and onto?
f1 = {(1, 3), (2, 5), (3, 7)} ; A = {1, 2, 3}, B = {3, 5, 7}

Ex. 1.5 | Q 2.2 | Page 31

Which of the following functions from A to B are one-one and onto?

f2 = {(2, a), (3, b), (4, c)} ; A = {2, 3, 4}, B = {abc}

Ex. 1.5 | Q 2.3 | Page 31

Which of the following functions from A to B are one-one and onto ?

f3 = {(ax), (bx), (cz), (dz)} ; A = {abcd,}, B = {xyz}.

Ex. 1.5 | Q 3 | Page 31

Prove that the function f : N → N, defined by f(x) = x2 + x + 1, is one-one but not onto

Ex. 1.5 | Q 4 | Page 31

Let A = {−1, 0, 1} and f = {(xx2) : x ∈ A}. Show that f : A → A is neither one-one nor onto.

Ex. 1.5 | Q 5.01 | Page 31

Classify the following function as injection, surjection or bijection : f : N → N given by f(x) = x2

Ex. 1.5 | Q 5.02 | Page 31

Classify the following function as injection, surjection or bijection :  f : Z → Z given by f(x) = x2

Ex. 1.5 | Q 5.03 | Page 31

Classify the following function as injection, surjection or bijection : f : N → N given by f(x) = x3

Ex. 1.5 | Q 5.04 | Page 31

Classify the following function as injection, surjection or bijection :  f : Z → Z given by f(x) = x3

Ex. 1.5 | Q 5.05 | Page 31

Classify the following function as injection, surjection or bijection :

f : R → R, defined by f(x) = |x|

Ex. 1.5 | Q 5.06 | Page 31

Classify the following function as injection, surjection or bijection :

f : Z → Z, defined by f(x) = x2 + x

Ex. 1.5 | Q 5.07 | Page 31

Classify the following function as injection, surjection or bijection :

f : Z → Z, defined by f(x) = x − 5

Ex. 1.5 | Q 5.08 | Page 31

Classify the following function as injection, surjection or bijection :

f : R → R, defined by f(x) = sinx

Ex. 1.5 | Q 5.09 | Page 31

Classify the following function as injection, surjection or bijection :

f : R → R, defined by f(x) = x3 + 1

Ex. 1.5 | Q 5.1 | Page 31

Classify the following function as injection, surjection or bijection :

f : R → R, defined by f(x) = x3 − x

Ex. 1.5 | Q 5.11 | Page 31

Classify the following function as injection, surjection or bijection :

f : R → R, defined by f(x) = sin2x + cos2x

Ex. 1.5 | Q 5.12 | Page 31

Classify the following function as injection, surjection or bijection :

f : Q − {3} → Q, defined by f (x) = (2x +3)/(x-3)

Ex. 1.5 | Q 5.13 | Page 31

Classify the following function as injection, surjection or bijection :

f : Q → Q, defined by f(x) = x3 + 1

Ex. 1.5 | Q 5.14 | Page 31

Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 5x3 + 4

Ex. 1.5 | Q 5.15 | Page 31

Classify the following function as injection, surjection or bijection :

f : R → R, defined by f(x) = 3 − 4x

Ex. 1.5 | Q 5.16 | Page 31

Classify the following function as injection, surjection or bijection :

f : R → R, defined by f(x) = 1 + x2

Ex. 1.5 | Q 5.17 | Page 31

Classify the following function as injection, surjection or bijection :

f : R → R, defined by f(x) = x/(x^2 +1)

Ex. 1.5 | Q 6 | Page 31

If f : A → B is an injection, such that range of f = {a}, determine the number of elements in A.

Ex. 1.5 | Q 7 | Page 31

Show that the function f : R − {3} → R − {2} given by f(x) = (x-2)/(x-3) is a bijection.

Ex. 1.5 | Q 8.1 | Page 32

Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : f (x) = x/2

Ex. 1.5 | Q 8.2 | Page 32

Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : g(x) = |x|

Ex. 1.5 | Q 8.3 | Page 32

Let A = [-1, 1]. Then, discuss whether the following functions from A to itself is one-one, onto or bijective : h(x) = x2

Ex. 1.5 | Q 9.1 | Page 32

Set of ordered pair of  a function? If so, examine whether the mapping is injective or surjective :{(xy) : x is a person, y is the mother of x}

Ex. 1.5 | Q 9.2 | Page 32

Set of ordered pair of a function ? If so, examine whether the mapping is injective or surjective :{(ab) : a is a person, b is an ancestor of a

Ex. 1.5 | Q 10 | Page 32

Let A = {1, 2, 3}. Write all one-one from A to itself.

Ex. 1.5 | Q 11 | Page 32

If f : R → R be the function defined by f(x) = 4x3 + 7, show that f is a bijection.

Ex. 1.5 | Q 12 | Page 32

Show that the exponential function f : R → R, given by f(x) = ex, is one-one but not onto. What happens if the co-domain is replaced byR0^+ (set of all positive real numbers)?

Ex. 1.5 | Q 13 | Page 32

Show that the logarithmic function  f : R0+ → R   given  by f (x)  loga x ,a> 0   is   a  bijection.

Ex. 1.5 | Q 14 | Page 32

If A = {1, 2, 3}, show that a one-one function f : A → A must be onto.

Ex. 1.5 | Q 15 | Page 32

If A = {1, 2, 3}, show that a onto function f : A → A must be one-one.

Ex. 1.5 | Q 16 | Page 32

Find the number of all onto functions from the set A = {1, 2, 3, ..., n} to itself.

Ex. 1.5 | Q 17 | Page 32

Give examples of two one-one functions f1 and f2 from R to R, such that f1 + f2 : R → R. defined by (f1 + f2) (x) = f1 (x) + f2 (x) is not one-one.

Ex. 1.5 | Q 18 | Page 32

Give examples of two surjective functions f1 and f2 from Z to Z such that f1 + f2 is not surjective.

Ex. 1.5 | Q 19 | Page 32

Show that if f1 and f2 are one-one maps from R to R, then the product f1 × f2 : R → R defined by (f1 × f2) (x) = f1 (x) f2 (x) need not be one - one.

Ex. 1.5 | Q 20 | Page 32

Suppose f1 and f2 are non-zero one-one functions from R to R. Is f_1 / f^2 necessarily one - one? Justify your answer. Here,f_1/f_2 : R → R   is   given   by   (f_1/f_2) (x) = (f_1(x))/(f_2 (x))  for all  x in R .

Ex. 1.5 | Q 21 | Page 32

Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of each of the following:
(i) an injective map from A to B
(ii) a mapping from A to B which is not injective
(iii) a mapping from A to B.

Ex. 1.5 | Q 22 | Page 32

Show that f : R→ R, given by f(x) = x — [x], is neither one-one nor onto.

Ex. 1.5 | Q 23 | Page 32

Let f : N → N be defined by

f(n) = { (n+ 1, if n  is  odd),( n-1 , if n  is  even):}

Show that f is a bijection.

[CBSE 2012, NCERT]

## Chapter 1: Relations

Ex. 1.1Ex. 1.2Ex. 1.3Ex. 1.4Ex. 1.5

## RD Sharma solutions for Class 12 Mathematics chapter 1 - Relations

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