RD Sharma solutions for Class 12 Maths chapter 1 - Relations [Latest edition]

Chapters Chapter 1: Relations

Exercise 1.1Exercise 1.2Exercise 1.3Exercise 1.4
Exercise 1.1 [Pages 10 - 11]

RD Sharma solutions for Class 12 Maths Chapter 1 RelationsExercise 1.1 [Pages 10 - 11]

Exercise 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 relation is reflexive, symmetric and transitive :

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

Exercise 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 relation is reflexive, symmetric and transitive :

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

Exercise 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 relation is reflexive, symmetric and transitive:

R = {(xy) : x is wife of y}

Exercise 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 relation is reflexive, symmetric and transitive:

R = {(xy) : is father of and y}

Exercise 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, R2, R3, R4 on A is (i) reflexive (ii) symmetric and (iii) transitive.

Exercise 1.1 | 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.

Exercise 1.1 | 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

Exercise 1.1 | 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.

Exercise 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.

Exercise 1.1 | 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.

Exercise 1.1 | 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.

Exercise 1.1 | 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.

Exercise 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.

Exercise 1.1 | Q 7 | Page 11

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

Exercise 1.1 | Q 8 | Page 11

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

Exercise 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 ?

Exercise 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 ?

Exercise 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 ?

Exercise 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.

Exercise 1.1 | Q 11 | Page 11

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

Exercise 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.

Exercise 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 ?

Exercise 1.1 | Q 14.1 | Page 11

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

Exercise 1.1 | Q 14.2 | Page 11

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

Exercise 1.1 | Q 14.3 | Page 11

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

Exercise 1.1 | Q 14.4 | Page 11

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

Exercise 1.1 | Q 14.5 | Page 11

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

Exercise 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.

Exercise 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.

Exercise 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.

Exercise 1.1 | Q 18.1 | Page 11

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

Determine the above relation is reflexive, symmetric and transitive.

Exercise 1.1 | Q 18.2 | Page 11

Defines a relation on :

x + y = 10, xy∈ N

Determine the above relation is reflexive, symmetric and transitive.

Exercise 1.1 | 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.

Exercise 1.1 | 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.

Exercise 1.2 [Pages 26 - 27]

RD Sharma solutions for Class 12 Maths Chapter 1 RelationsExercise 1.2 [Pages 26 - 27]

Exercise 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.

Exercise 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.

Exercise 1.2 | 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.

Exercise 1.2 | 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.

Exercise 1.2 | 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.

Exercise 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?

Exercise 1.2 | 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.

Exercise 1.2 | 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.

Exercise 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 = {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.

Exercise 1.2 | 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?

Exercise 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.

Exercise 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 = {(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}.

Exercise 1.2 | 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.

Exercise 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 (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.

Exercise 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 ?

Exercise 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 ?

Exercise 1.2 | 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.

Exercise 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.

Exercise 1.3 [Pages 29 - 30]

RD Sharma solutions for Class 12 Maths Chapter 1 RelationsExercise 1.3 [Pages 29 - 30]

Exercise 1.3 | 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

Exercise 1.3 | Q 2 | Page 30

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

Exercise 1.3 | Q 3 | Page 30

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

Exercise 1.3 | Q 4 | Page 30

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

Exercise 1.3 | Q 5 | Page 30

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

Exercise 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.

Exercise 1.3 | 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.

Exercise 1.3 | 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.

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

Exercise 1.3 | Q 10 | Page 30

Define a reflexive relation ?

Exercise 1.3 | Q 11 | Page 30

Define a symmetric relation ?

Exercise 1.3 | Q 12 | Page 30

Define a transitive relation ?

Exercise 1.3 | Q 13 | Page 30

Define an equivalence relation ?

Exercise 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.

Exercise 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.

Exercise 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.

Exercise 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 ?

Exercise 1.3 | Q 18 | Page 30

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

Exercise 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 .

Exercise 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.

Exercise 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?

Exercise 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.

Exercise 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

Exercise 1.3 | Q 24 | Page 30

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

Exercise 1.4 [Pages 31 - 33]

RD Sharma solutions for Class 12 Maths Chapter 1 RelationsExercise 1.4 [Pages 31 - 33]

Exercise 1.4 | 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

Exercise 1.4 | 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

Exercise 1.4 | 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

Exercise 1.4 | 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

Exercise 1.4 | 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

Exercise 1.4 | 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

Exercise 1.4 | 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

Exercise 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 ________________ .

• 1

• 2

• 3

• 4

Exercise 1.4 | 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

Exercise 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 ______________ .

• {1, 4, 6, 9}

• {4, 6, 9}

• {1}

• none of these

Exercise 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 ______________ .

• {2, 3, 5}

• {3, 5}

• {2, 3, 4}

• {2, 3, 4, 5}

Exercise 1.4 | 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

Exercise 1.4 | 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}

Exercise 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 ______________ .

• {(8, 11), (10, 13)}

• {(11, 8), (13, 10)}

• {(10, 13), (8, 11)}

• none of these

Exercise 1.4 | 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

Exercise 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 ________________ .

• neither reflexive nor transitive

• neither symmetric nor transitive

• transitive

• none of these

Exercise 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 _____________ .

• R ⊂ S

• S ⊂ R

• R = S

• none of these

Exercise 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 = _____________ .

• {(3, 1), (6, 2), (8, 2), (9, 3)}

• {(3, 1), (6, 2), (9, 3)}

• {(3, 1), (2, 6), (3, 9)}

• none of these

Exercise 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 ____________ .

• reflexive

• symmetric

• transitive

• all the three options

Exercise 1.4 | 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

Exercise 1.4 | 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

Exercise 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, _____________________ .

• 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

Exercise 1.4 | Q 23 | Page 32

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

• 1

• 2

• 3

• 4

Exercise 1.4 | 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

Exercise 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

Exercise 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)

Exercise 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

Exercise 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

Exercise 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

Exercise 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

Exercise 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

Exercise 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

Exercise 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

Exercise 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

Exercise 1.1Exercise 1.2Exercise 1.3Exercise 1.4 RD Sharma solutions for Class 12 Maths 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 Class 12 Maths solutions in a manner that help students grasp basic concepts better and faster.

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

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