#### Chapters

Chapter 2: Relations

Chapter 3: Functions

Chapter 4: Measurement of Angles

Chapter 5: Trigonometric Functions

Chapter 6: Graphs of Trigonometric Functions

Chapter 7: Values of Trigonometric function at sum or difference of angles

Chapter 8: Transformation formulae

Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle

Chapter 10: Sine and cosine formulae and their applications

Chapter 11: Trigonometric equations

Chapter 12: Mathematical Induction

Chapter 13: Complex Numbers

Chapter 14: Quadratic Equations

Chapter 15: Linear Inequations

Chapter 16: Permutations

Chapter 17: Combinations

Chapter 18: Binomial Theorem

Chapter 19: Arithmetic Progression

Chapter 20: Geometric Progression

Chapter 21: Some special series

Chapter 22: Brief review of cartesian system of rectangular co-ordinates

Chapter 23: The straight lines

Chapter 24: The circle

Chapter 25: Parabola

Chapter 26: Ellipse

Chapter 27: Hyperbola

Chapter 28: Introduction to three dimensional coordinate geometry

Chapter 29: Limits

Chapter 30: Derivatives

Chapter 31: Mathematical reasoning

Chapter 32: Statistics

Chapter 33: Probability

#### RD Sharma Mathematics Class 11

## Chapter 2: Relations

#### Chapter 2: Relations Exercise 2.10 solutions [Pages 2 - 8]

(i) If \[\left( \frac{a}{3} + 1, b - \frac{2}{3} \right) = \left( \frac{5}{3}, \frac{1}{3} \right)\] find the values of *a* and *b*.

(ii) If (*x* + 1, 1) = (3, *y* − 2), find the values of *x* and *y*.

If the ordered pairs (*x*, −1) and (5, *y*) belong to the set {(*a*, *b*) : *b* = 2*a* − 3}, find the values of *x* and *y*.

If *a* ∈ [−1, 2, 3, 4, 5] and *b* ∈ [0, 3, 6], write the set of all ordered pairs (*a*, *b*) such that *a* + *b*= 5.

If *a* ∈ [2, 4, 6, 9] and *b* ∈ [4, 6, 18, 27], then form the set of all ordered pairs (*a*, *b*) such that a divides *b* and *a* < *b*.

If *A* = {1, 2} and *B* = {1, 3}, find *A* × *B* and *B* × *A*.

Let *A* = {1, 2, 3} and *B* = {3, 4}. Find *A* × *B* and show it graphically.

If *A* = {1, 2, 3} and *B* = {2, 4}, what are *A* × *B*, *B* × *A*, *A* × *A*, *B* × *B* and (*A* × *B*) ∩ (*B* × *A*)?

If *A* and *B* are two set having 3 elements in common. If *n*(*A*) = 5, *n*(*B*) = 4, find *n*(*A* × *B*) and *n*[(*A* × *B*) ∩ (*B* × *A*)].

Let *A* and *B* be two sets. Show that the sets *A* × *B* and *B* × *A* have elements in common iff the sets *A* and *B* have an elements in common.

Let *A* and *B* be two sets such that *n*(*A*) = 3 and *n*(*B*) = 2.

If (*x*, 1), (*y*, 2), (*z*, 1) are in *A* × *B*, find *A* and *B*, where *x*, *y*, *z* are distinct elements.

Let *A* = {1, 2, 3, 4} and R = {(*a*, *b*) : *a* ∈ *A*, *b* ∈ *A*, *a* divides *b*}. Write *R* explicitly.

If *A* = {−1, 1}, find *A* × *A* × *A*.

State whether of the statement is true or false. If the statement is false, re-write the given statement correctly:

If P = {m, n} and Q = {n, m}, then P × Q = {(m, n), (n, m)}

True

False

State whether of the statement is true or false. If the statement is false, re-write the given statement correctly:

If A and B are non-empty sets, then A × B is a non-empty set of ordered pairs (x, y) such that x ∈ B and y ∈ A.

True

False

State whether of the statement is true or false. If the statement is false, re-write the given statement correctly:

(iii) If A = {1, 2}, B = {3, 4}, then A × (B ∩ ϕ) = ϕ.

True

False

If *A* = {1, 2}, from the set *A* × *A* × *A*.

If A = {1, 2, 4} and B = {1, 2, 3}, represent set graphically:

(i) A × B

If A = {1, 2, 4} and B = {1, 2, 3}, represent set graphically:

(ii) B × A

If A = {1, 2, 4} and B = {1, 2, 3}, represent set graphically:

(iii) A × A

If A = {1, 2, 4} and B = {1, 2, 3}, represent set graphically:

(iv) B × B

#### Chapter 2: Relations Exercise 2.20 solutions [Page 12]

Given A = {1, 2, 3}, B = {3, 4}, C ={4, 5, 6}, find (A × B) ∩ (B × C ).

If *A* = {2, 3}, *B* = {4, 5}, *C* ={5, 6}, find *A* × (*B* ∪ *C*), A × (*B* ∩ *C*), (*A* × *B*) ∪ (*A* × *C*).

If *A* = {1, 2, 3}, *B* = {4}, *C* = {5}, then verify that:

(i) *A* × (*B* ∪ *C*) = (*A* × *B*) ∪ (*A* × *C*)

If *A* = {1, 2, 3}, *B* = {4}, *C* = {5}, then verify that:

(ii) *A* × (*B* ∩ *C*) = (*A* × *B*) ∩ (*A* × *C*)

If *A* = {1, 2, 3}, *B* = {4}, *C* = {5}, then verify that:

(iii) *A* × (*B* − *C*) = (*A* × *B*) − (*A* × *C*)

Let *A* = {1, 2}, *B* = {1, 2, 3, 4}, *C* = {5, 6} and *D* = {5, 6, 7, 8}. Verify that:

(i) *A* × *C* ⊂ *B* × *D*

Let *A* = {1, 2}, *B* = {1, 2, 3, 4}, *C* = {5, 6} and *D* = {5, 6, 7, 8}. Verify that:

(ii) *A* × (*B* ∩ *C*) = (*A* × *B*) ∩ (*A* × *C*)

If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find

(i) A × (B ∩ C)

If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find

(ii) (A × B) ∩ (A × C)

If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find

(iii) A × (B ∪ C)

If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find

(iv) (A × B) ∪ (A × C)

Prove that:

(i) (*A* ∪ *B*) × *C* = (*A* × *C*) ∪ (*B* × *C*)

(ii) (*A* ∩ B) × C = (*A* × *C*) ∩ (B×*C*)

If *A* × *B* ⊆ *C* × *D* and *A* × *B* ≠ ϕ, prove that *A* ⊆ *C* and *B* ⊆ *D*.

#### Chapter 2: Relations Exercise 2.30 solutions [Pages 20 - 21]

If A = [1, 2, 3], B = [4, 5, 6], which of the following are relations from A to B? Give reasons in support of your answer.

(i) [(1, 6), (3, 4), (5, 2)]

(ii) [(1, 5), (2, 6), (3, 4), (3, 6)]

(iii) [(4, 2), (4, 3), (5, 1)]

(iv) A × B.

A relation R is defined from a set A = [2, 3, 4, 5] to a set B = [3, 6, 7, 10] as follows:

(*x*, *y*) ∈ R ⇔ *x* is relatively prime to *y*

Express R as a set of ordered pairs and determine its domain and range.

Let A be the set of first five natural numbers and let R be a relation on A defined as follows:

(*x*, *y*) ∈ R ⇔ *x* ≤ *y*

Express R and R^{−1} as sets of ordered pairs. Determine also (i) the domain of R^{−1} (ii) the range of R.

Find the inverse relation R^{−1} in each of the cases:

(i) R = {(1, 2), (1, 3), (2, 3), (3, 2), (5, 6)}

Find the inverse relation R^{−1} in each of the cases:

(ii) R = {(*x*, *y*), : *x*, *y* ∈ N, *x* + 2*y* = 8}

Find the inverse relation R^{−1} in each of the cases:

(iii) R is a relation from {11, 12, 13} to (8, 10, 12] defined by *y* = *x* − 3.

Write the relation as the sets of ordered pairs:

(i) A relation R from the set [2, 3, 4, 5, 6] to the set [1, 2, 3] defined by *x* = 2*y*.

Write the relation as the sets of ordered pairs:

(ii) A relation R on the set [1, 2, 3, 4, 5, 6, 7] defined by (*x*, *y*) ∈ R ⇔ *x* is relatively prime to *y*.

Write the relation as the sets of ordered pairs:

(iii) A relation R on the set [0, 1, 2, ....., 10] defined by 2*x* + 3*y* = 12.

Write the relation as the sets of ordered pairs:

(iv) A relation R from a set A = [5, 6, 7, 8] to the set B = [10, 12, 15, 16,18] defined by (*x*, *y*) ∈ R ⇔ *x* divides *y*.

Let *R* be a relation in N defined by (*x*, *y*) ∈ R ⇔ *x* + 2*y* =8. Express R and R^{−1} as sets of ordered pairs.

Let A = (3, 5) and B = (7, 11). Let R = {(*a*, *b*) : *a* ∈ A, *b* ∈ B, *a* − *b* is odd}. Show that R is an empty relation from A into B.

Let A = [1, 2] and B = [3, 4]. Find the total number of relation from A into B.

Determine the domain and range of the relation R defined by

(i) R = [(*x*, *x* + 5): *x* ∈ (0, 1, 2, 3, 4, 5)]

Determine the domain and range of the relation R defined by

(ii) R = {(*x*, *x*^{3}) :* x* is a prime number less than 10}

Determine the domain and range of the relations:

(i) R = {(*a*, *b*) : *a* ∈ N, *a* < 5, *b* = 4}

Determine the domain and range of the relations:

(ii) \[S = \left\{ \left( a, b \right) : b = \left| a - 1 \right|, a \in Z \text{ and} \left| a \right| \leq 3 \right\}\]

Let A = {*a*, *b*}. List all relations on A and find their number.

Let A = (*x*, *y*, *z*) and B = (*a*, *b*). Find the total number of relations from A into B.

Let R be a relation from N to N defined by R = [(*a*, *b*) : *a*, *b* ∈ N and *a* = *b*^{2}]. Are the statement true?

(i) (*a*, *a*) ∈ R for all *a* ∈ N

Let R be a relation from N to N defined by R = [(*a*, *b*) : *a*, *b* ∈ N and *a* = *b*^{2}]. Are the statement true?

(ii) (*a*, *b*) ∈ R ⇒ (*b*, *a*) ∈ R

Let R be a relation from N to N defined by R = [(*a*, *b*) : *a*, *b* ∈ N and *a* = *b*^{2}]. Are the statement true?

(iii) (*a*, *b*) ∈ R and (*b*, *c*) ∈ R ⇒ (*a*, *c*) ∈ R

Let A = [1, 2, 3, ......., 14]. Define a relation on a set A by

R = {(*x*, *y*) : 3*x* − *y* = 0, where *x*, *y* ∈ A}.

Depict this relationship using an arrow diagram. Write down its domain, co-domain and range.

Define a relation R on the set N of natural number by R = {(*x*, *y*) : *y* = *x* + 5, *x* is a natural number less than 4, *x*, *y* ∈ N}. Depict this relationship using (i) roster form (ii) an arrow diagram. Write down the domain and range or R.

A = [1, 2, 3, 5] and B = [4, 6, 9]. Define a relation R from A to B by R = {(*x*, *y*) : the difference between *x* and *y* is odd, *x* ∈ A, *y* ∈ B}. Write R in Roster form.

Write the relation R = {(*x*, *x*^{3}) : *x* is a prime number less than 10} in roster form.

Let A = [1, 2, 3, 4, 5, 6]. Let R be a relation on A defined by {(*a*, *b*) : *a*, *b* ∈ A, *b* is exactly divisible by *a*}

(i) Writer R in roster form

(ii) Find the domain of R

(ii) Find the range of R.

The adjacent figure shows a relationship between the sets P and Q. Write this relation in (i) set builder form (ii) roster form. What is its domain and range?

Let R be the relation on Z defined by

R = {(*a*, *b*) : *a*, *b* ∈ Z, *a* − *b* is an integer}

Find the domain and range of R.

For the relation R_{1} defined on R by the rule (*a*, *b*) ∈ R_{1} ⇔ 1 + *ab* > 0. Prove that: (*a*, *b*) ∈ R_{1} and (*b* , *c*) ∈ R_{1} ⇒ (*a*, *c*) ∈ R_{1} is not true for all *a*, *b*, *c* ∈ R.

Let R be a relation on N × N defined by

(*a*, *b*) R (*c*, *d*) ⇔ *a* + *d* = *b* + *c* for all (*a*, *b*), (*c*, *d*) ∈ N × N

Show that:

(i) (*a*, *b*) R (*a*, *b*) for all (*a*, *b*) ∈ N × N

Let R be a relation on N × N defined by

(*a*, *b*) R (*c*, *d*) ⇔ *a* + *d* = *b* + *c* for all (*a*, *b*), (*c*, *d*) ∈ N × N

Show that:

(ii) (*a*, *b*) R (*c*, *d*) ⇒ (*c*, *d*) R (*a*, *b*) for all (*a*, *b*), (*c*, *d*) ∈ N × N

Let R be a relation on N × N defined by

(*a*, *b*) R (*c*, *d*) ⇔ *a* + *d* = *b* + *c* for all (*a*, *b*), (*c*, *d*) ∈ N × N

(iii) (*a*, *b*) R (*c*, *d*) and (*c*, *d*) R (*e*, *f*) ⇒ (*a*, *b*) R (*e*, *f*) for all (*a*, *b*), (*c*, *d*), (*e*, *f*) ∈ N × N

#### Chapter 2: Relations solutions [Pages 24 - 25]

If A = {1, 2, 4}, B = {2, 4, 5} and C = {2, 5}, write (A − C) × (B − C).

If *n*(A) = 3, *n*(B) = 4, then write *n*(A × A × B).

If R is a relation defined on the set Z of integers by the rule (*x*, *y*) ∈ R ⇔ *x*^{2} + *y*^{2} = 9, then write domain of R.

If R = {(*x*, *y*) : *x*, *y* ∈ Z, *x*^{2} + *y*^{2} ≤ 4} is a relation defined on the set Z of integers, then write domain of R.

If R is a relation from set A = (11, 12, 13) to set B = (8, 10, 12) defined by *y* = *x* − 3, then write R^{−1}.

Let A = {1, 2, 3} and\[R = \left\{ \left( a, b \right) : \left| a^2 - b^2 \right| \leq 5, a, b \in A \right\}\].Then write R as set of ordered pairs.

Let R = [(*x*, *y*) : *x*, *y* ∈ Z, *y* = 2*x* − 4]. If (*a*, -2) and (4, *b*^{2}) ∈ R, then write the values of *a* and *b*.

If R = {(2, 1), (4, 7), (1, −2), ...}, then write the linear relation between the components of the ordered pairs of the relation R.

If A = [1, 3, 5] and B = [2, 4], list of elements of R, if

R = {(*x*, *y*) : *x*, *y* ∈ A × B and *x* > *y*}

If R = [(*x*, *y*) : *x*, *y* ∈ W, 2*x* + *y* = 8], then write the domain and range of R.

Let A and B be two sets such that *n*(A) = 3 and *n*(B) = 2. If (*x*, 1), (*y*, 2), (*z*, 1) are in A × B, write A and B

Let A = [1, 2, 3, 5], B = [4, 6, 9] and R be a relation from A to B defined by R = {(*x*, *y*) : *x* − *y*is odd}. Write R in roster form.

#### Chapter 2: Relations solutions [Pages 25 - 26]

If A = {1, 2, 4}, B = {2, 4, 5}, C = {2, 5}, then (A − B) × (B − C) is

(a) {(1, 2), (1, 5), (2, 5)}

(b) [(1, 4)]

(c) (1, 4)

(d) none of these

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

Let A = [1, 2, 3], B = [1, 3, 5]. If relation R from A to B is given by = {(1, 3), (2, 5), (3, 3)}, Then R^{−1} is

(a) {(3, 3), (3, 1), (5, 2)}

(b) {(1, 3), (2, 5), (3, 3)}

(c) {(1, 3), (5, 2)}

(d) None of these

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.

If R = {(*x*, *y*) : *x*, *y* ∈ Z, *x*^{2} +* **y*^{2} ≤ 4} is a relation on Z, then the domain of R is

(a) [0, 1, 2]

(b) [0, −1, −2]

(c) {−2, −1, 0, 1, 2]

(d) None of these

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]

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

(a) (2 + 3

*i*) ϕ 13(b) 3ϕ (−3)

(c) (1 +

*i*) ϕ 2(d)

*i*ϕ 1

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

(a) [2, 4, 8]

(b) [2, 4, 6, 8]

(c) [2, 4, 6]

(d) [1, 2, 3, 4]

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), (12, 10)]

(d) none of these

If the set A has *p* elements, B has *q* elements, then the number of elements in A × B is

(a) p + q

(b)

*p*+*q*+ 1(c) pq

(d)

*p*^{2}

Let R be a relation from a set A to a set B, then

(a) R = A ∪ B

(b) R = A ∩ B

(c) R ⊆ A × B

(d) R ⊆ B × A

If R is a relation from a finite set A having *m* elements of a finite set B having *n* elements, then the number of relations from A to B is

(a) 2

^{mn}(b) 2

^{mn}− 1(c) 2

*mn*(d)

*m*^{n}

If R is a relation on a finite set having *n* elements, then the number of relations on A is

(a) 2

^{n}(b) \[2^{n^2}\]

(c)

*n*^{2}(d)

*n*^{n}

## Chapter 2: Relations

#### RD Sharma Mathematics Class 11

#### Textbook solutions for Class 11

## RD Sharma solutions for Class 11 Mathematics chapter 2 - Relations

RD Sharma solutions for Class 11 Maths chapter 2 (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 Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics chapter 2 Relations are Cartesian Product of Sets, Brief Review of Cartesian System of Rectanglar Co-ordinates, Relation, Functions, Some Functions and Their Graphs, Algebra of Real Functions, Ordered Pairs, Equality of Ordered Pairs, Pictorial Diagrams, Graph of Function, Pictorial Representation of a Function, Exponential Function, Logarithmic Functions.

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