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RD Sharma solutions for Class 12 Mathematics chapter 1 - Relations

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

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RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Chapter 1: Relations

Ex. 1.10Ex. 1.20Ex. 1.30Ex. 1.40Ex. 1.4

Chapter 1: Relations Exercise 1.10 solutions [Pages 10 - 11]

Ex. 1.10 | 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 relation is reflexive, symmetric and transitive :

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

Ex. 1.10 | 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 relation is reflexive, symmetric and transitive :

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

Ex. 1.10 | 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 relation is reflexive, symmetric and transitive:

R = {(xy) : x is wife of y}

Ex. 1.10 | 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 relation is reflexive, symmetric and transitive:

R = {(xy) : is father of and y}

Ex. 1.10 | 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. 1.10 | Q 3.1 | Page 10

Test whether the following relation R1 is  (i) reflexive (ii) symmetric and (iii) transitive :

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

Ex. 1.10 | Q 3.2 | Page 10

Test whether the following relation R2 is (i) reflexive (ii) symmetric and (iii) transitive:

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

Ex. 1.10 | Q 3.3 | Page 10

Test whether the following relation R3 is (i) reflexive (ii) symmetric and (iii) transitive:

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

Ex. 1.10 | 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. 1.10 | Q 5.1 | Page 11

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

Find whether relation is reflexive, symmetric or transitive.

Ex. 1.10 | Q 5.2 | Page 11

The following relation is defined on the set of real numbers.

aRb if 1 + ab > 0

Find whether relation is reflexive, symmetric or transitive.

Ex. 1.10 | Q 5.3 | Page 11

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

Find whether relation is reflexive, symmetric or transitive.

Ex. 1.10 | 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. 1.10 | Q 7 | Page 11

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

Ex. 1.10 | Q 8 | Page 11

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

Ex. 1.10 | 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. 1.10 | 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. 1.10 | Q 9.3 | Page 11

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

Ex. 1.10 | 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. 1.10 | Q 11 | Page 11

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

Ex. 1.10 | 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. 1.10 | Q 13 | Page 11

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

Ex. 1.10 | Q 14.1 | Page 11

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

Ex. 1.10 | Q 14.2 | Page 11

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

Ex. 1.10 | Q 14.3 | Page 11

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

Ex. 1.10 | Q 14.4 | Page 11

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

Ex. 1.10 | Q 14.5 | Page 11

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

Ex. 1.10 | 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. 1.10 | 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. 1.10 | 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.

Ex. 1.10 | Q 18.1 | Page 11

Defines a relation on :
  x > y, x, y ∈  N

Determine the above relation is reflexive, symmetric and transitive.

Ex. 1.10 | Q 18.2 | Page 11

Defines a relation on :

x + y = 10, xy∈ N

Determine the above relation is reflexive, symmetric and transitive.

Ex. 1.10 | Q 18.3 | Page 11

Defines a relation on N :

xy is square of an integer, x, y ∈ N

Determine the above relation is reflexive, symmetric and transitive.

Ex. 1.10 | Q 18.4 | Page 11

Defines a relation on N:

x + 4y = 10, x, y ∈ N

Determine the above relation is reflexive, symmetric and transitive.

Chapter 1: Relations Exercise 1.20 solutions [Pages 26 - 27]

Ex. 1.20 | 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. 1.20 | 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. 1.20 | Q 3 | Page 26

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

Ex. 1.20 | Q 4 | Page 26

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

Ex. 1.20 | Q 5 | Page 26

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

Ex. 1.20 | 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. 1.20 | Q 7 | Page 26

Let R be a relation on the set A of ordered pair of integers defined by (x, y) R (u, v) if xv = yu. Show that R is an equivalence relation.

Ex. 1.20 | Q 8 | Page 26

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

Ex. 1.20 | 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 = {L1, L2) : 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. 1.20 | Q 10 | Page 27

Show that the relation R, defined on the set A of all polygons as
R = {(P1, P2) : 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. 1.20 | 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. 1.20 | Q 12 | Page 27

Let R be the relation defined on the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b) : 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. 1.20 | Q 13 | Page 27

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

Ex. 1.20 | 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 (a, b) R (c, d) ⇔ ad = bc for all (a, b), (c, d) ∈ Z × Z0,
Prove that R is an equivalence relation on Z × Z0.

Ex. 1.20 | 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. 1.20 | 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. 1.20 | Q 16 | Page 27

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

Ex. 1.20 | 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 1.30 solutions [Pages 29 - 30]

Ex. 1.30 | Q 1 | Page 29

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

Ex. 1.30 | Q 2 | Page 30

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

Ex. 1.30 | Q 3 | Page 30

Write the identity relation on set A = {a, b, c}.

Ex. 1.30 | Q 4 | Page 30

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

Ex. 1.30 | Q 5 | Page 30

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

Ex. 1.30 | Q 6 | Page 30

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

Ex. 1.30 | Q 7 | Page 30

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

Ex. 1.30 | Q 8 | Page 30

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

Ex. 1.30 | 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 = {(x, y) : x and y are relatively prime}. Then, write R and R−1.

Ex. 1.30 | Q 10 | Page 30

Define a reflexive relation ?

Ex. 1.30 | Q 11 | Page 30

Define a symmetric relation ?

Ex. 1.30 | Q 12 | Page 30

Define a transitive relation ?

Ex. 1.30 | Q 13 | Page 30

Define an equivalence relation ?

Ex. 1.30 | 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. 1.30 | 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. 1.30 | 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. 1.30 | 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. 1.30 | Q 18 | Page 30

Let R = {(a, a3) : a is a prime number less than 5} be a relation. Find the range of R.

Ex. 1.30 | 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. 1.30 | 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. 1.30 | 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. 1.30 | 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. 1.30 | 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. 1.30 | Q 24 | Page 30

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

Chapter 1: Relations Exercise 1.40, 1.4 solutions [Pages 31 - 33]

Ex. 1.40 | Q 1 | Page 31

Let R be a relation on the set N given by
R = {(a, b) : a = b − 2, b > 6}. Then,

  • (2, 4) ∈ R

  • (3, 8) ∈ R

  • (6, 8) ∈ R

  • (8, 7) ∈ R

Ex. 1.40 | Q 2 | Page 31

If a relation R is defined on the set Z of integers as follows:
(a, b) ∈ R ⇔ a2 + b2 = 25. Then, domain (R) is ___________

  • {3, 4, 5}

  • {0, 3, 4, 5}

  • {0, ±3, ±4, ±5}

  • None of these

Ex. 1.40 | Q 3 | Page 31

R is a relation on the set Z of integers and it is given by
(x, y) ∈ R ⇔ | x − y | ≤ 1. Then, R is ______________ .

  • Reflexive and transitive

  • Reflexive and symmetric

  • Symmetric and transitive

  • an equivalence relation

Ex. 1.40 | Q 4 | Page 31

The relation R defined on the set A = {1, 2, 3, 4, 5} by
R = {(a, b) : | a2 − b2 | < 16} is given by ______________ .

  • {(1, 1), (2, 1), (3, 1), (4, 1), (2, 3)}

  • {(2, 2), (3, 2), (4, 2), (2, 4)}

  • {(3, 3), (4, 3), (5, 4), (3, 4)}

  • none of these

Ex. 1.40 | Q 5 | Page 31

Let R be the relation over the set of all straight lines in a plane such that  l1 R l2 ⇔ l 1⊥ l2. Then, R is _____________ .

  • Symmetric

  • Reflexive

  • Transitive

  • an equivalence relation

Ex. 1.40 | Q 6 | Page 31

If A = {a, b, c}, then the relation R = {(b, c)} on A is _______________ .

  • reflexive only

  • symmetric only

  • transitive only

  • reflexive and transitive only

Ex. 1.40 | Q 7 | Page 31

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

  • 4

  • 5

  • 6

  • 7

Ex. 1.40 | 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 ________________ .

  • 1

  • 2

  • 3

  • 4

Ex. 1.40 | Q 9 | Page 31

The relation 'R' in N × N such that
(a, b) R (c, d) ⇔ a + d = b + c is ______________ .

  • reflexive but not symmetric

  • reflexive and transitive but not symmetric

  • an equivalence relation

  • none of the these

Ex. 1.40 | 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 ______________ .

  • {1, 4, 6, 9}

  • {4, 6, 9}

  • {1}

  • none of these

Ex. 1.40 | 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 ______________ .

  • {2, 3, 5}

  • {3, 5}

  • {2, 3, 4}

  • {2, 3, 4, 5}

Ex. 1.40 | Q 12 | Page 32

A relation ϕ from C to R is defined by x ϕ y ⇔ | x | = y. Which one is correct?

  • (2 + 3 i) ϕ 13

  • 3 ϕ (−3)

  • (1 + i) ϕ 2

  • i ϕ 1

Ex. 1.40 | Q 13 | Page 32

Let R be a relation on N defined by x + 2y = 8. The domain of R is _______________ .

  • {2, 4, 8}

  • {2, 4, 6, 8}

  • {2, 4, 6}

  • {1, 2, 3, 4}

Ex. 1.40 | 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 ______________ .

  • {(8, 11), (10, 13)}

  • {(11, 8), (13, 10)}

  • {(10, 13), (8, 11)}

  • none of these

Ex. 1.40 | Q 15 | Page 32

Let R = {(a, a), (b, b), (c, c), (a, b)} be a relation on set A = a, b, c. Then, R is _______________ .

  • identify relation

  • reflexive

  • symmetric

  • antisymmetric

Ex. 1.40 | 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 ________________ .

  • neither reflexive nor transitive

  • neither symmetric nor transitive

  • transitive

  • none of these

Ex. 1.40 | Q 17 | Page 32

If R is the largest equivalence relation on a set A and S is any relation on A, then _____________ .

  • R ⊂ S

  • S ⊂ R

  • R = S

  • none of these

Ex. 1.40 | 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 = _____________ .

  • {(3, 1), (6, 2), (8, 2), (9, 3)}

  • {(3, 1), (6, 2), (9, 3)}

  • {(3, 1), (2, 6), (3, 9)}

  • none of these

Ex. 1.40 | 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 ____________ .

  • reflexive

  • symmetric

  • transitive

  • all the three options

Ex. 1.40 | Q 20 | Page 32

 If A = {a, b, c, d}, then a relation R = {(a, b), (b, a), (a, a)} on A is _____________ .

  • symmetric and transitive only

  • reflexive and transitive only

  • symmetric only

  • transitive only

Ex. 1.40 | Q 21 | Page 32

If A = {1, 2, 3}, then a relation R = {(2, 3)} on A is _____________ .

  • symmetric and transitive only

  • symmetric only

  • transitive only

  • none of these

Ex. 1.40 | 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, _____________________ .

  • R is reflexive and symmetric but not transitive

  • R is reflexive and transitive but not symmetric

  • R is symmetric and transitive but not reflexive

  • R is an equivalence relation

Ex. 1.40 | Q 23 | Page 32

Let A = {1, 2, 3}. Then, the number of equivalence relations containing (1, 2) is _______________ .

  • 1

  • 2

  • 3

  • 4

Ex. 1.40 | Q 24 | Page 32

The relation R = {(1, 1), (2, 2), (3, 3)} on the set {1, 2, 3} is ___________________ .

  • symmetric only

  • reflexive only

  • an equivalence relation

  • transitive only

Ex. 1.4 | Q 25 | Page 33

S is a relation over the set R of all real numbers and it is given by (a, b) ∈ S ⇔ ab ≥ 0. Then, S is _______________ .

  • symmetric and transitive only

  • reflexive and symmetric only

  • antisymmetric relation

  • an equivalence relation

Ex. 1.4 | Q 26 | Page 33

In the set Z of all integers, which of the following relation R is not an equivalence relation ?

  • x R y : if x ≤ y

  • x R y : if x = y

  • x R y : if x − y is an even integer

  • x R y : if x ≡ y (mod 3)

Ex. 1.4 | Q 27 | Page 33

Mark the correct alternative in the following question:

Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}. Then, R is _______________ .

  • reflexive but not symmetric

  • reflexive but not transitive

  • symmetric and transitive

  • neither symmetric nor transitive

Ex. 1.4 | Q 28 | Page 33

Mark the correct alternative in the following question:

The relation S defined on the set R of all real number by the rule aSb if a  b is _______________ .

  • an equivalence relation

  • reflexive, transitive but not symmetric

  • symmetric, transitive but not reflexive

  • neither transitive nor reflexive but symmetric

Ex. 1.4 | Q 29 | Page 33

Mark the correct alternative in the following question:

The maximum number of equivalence relations on the set A = {1, 2, 3} is _______________ .

  • 1

  • 2

  • 3

  • 5

Ex. 1.4 | Q 30 | Page 33

Mark the correct alternative in the following question:

Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then, R is _____________ .

  • Reflexive and symmetric

  • Transitive and symmetric

  • Equivalence

  • Reflexive, transitive but not symmetric

Ex. 1.4 | Q 31 | Page 33

Mark the correct alternative in the following question:

Let L denote the set of all straight lines in a plane. Let a relation R be defined by lRm if l is perpendicular to m for all l, m  L. Then, R is ______________ .

  • reflexive

  • symmetric

  • transitive

  • none of these

Ex. 1.4 | Q 32 | Page 33

Mark the correct alternative in the following question:

Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b for all a, b  T. Then, R is ____________ .

  • reflexive but not symmetric

  • transitive but not symmetric

  • equivalence

  • none of these

Ex. 1.4 | Q 33 | Page 33

Mark the correct alternative in the following question:

Consider a non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. Then, R is _____________ .

  • symmetric but not transitive

  • transitive but not symmetric

  • neither symmetric nor transitive

  • both symmetric and transitive

Ex. 1.4 | Q 34 | Page 33

Mark the correct alternative in the following question:

For real numbers x and y, define xRy if `x-y+sqrt2` is an irrational number. Then the relation R is ___________ .

  • reflexive

  • symmetric

  • transitive

  • none of these

Chapter 1: Relations

Ex. 1.10Ex. 1.20Ex. 1.30Ex. 1.40Ex. 1.4

RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

RD Sharma solutions for Class 12 Mathematics chapter 1 - Relations

RD Sharma solutions for Class 12 Maths chapter 1 (Relations) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 12 Mathematics chapter 1 Relations are Types of Relations, Types of Functions, Composition of Functions and Invertible Function, Inverse of a Function, Concept of Binary Operations, Introduction of Relations and Functions.

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