RD Sharma Class 11 Solutions Chapter - Relations | Mathematics (Maths) Class 11 - Commerce PDF Download

 Page 1


2. Relations
Exercise 2.1
1 A. Question
If , find the values of a and b.
Answer
Given: 
To find: values of a and b
By the definition of equality of ordered pairs, we have and simultaneously solving for a and b
and 
? b = 1
? a = 2
1 B. Question
If (x + 1, 1) = (3y, y - 1), find the values of x and y.
Answer
given: (x + 1, 1) = (3y, y - 1)
To find: values of a and b
By the definition of equality of ordered pairs, we have
x + 1 = 3y and  1 = y - 1
? x = 3y - 1 and  y = 2So,x = 3(2) - 1  = 6 - 1  = 5
? x = 5 and y = 2
2. Question
If the ordered pairs (x, - 1) and (5, y) belong to the set {(a, b): b = 2a - 3}, find the values of x and y.
Answer
given the ordered pairs (x, - 1) and (5, y) belong to the set {(a, b): b = 2a - 3}
To find: values of x and y
solving for first order pair
 (x, - 1) = {(a, b): b = 2a - 3}
 x = a and b = - 1
Page 2


2. Relations
Exercise 2.1
1 A. Question
If , find the values of a and b.
Answer
Given: 
To find: values of a and b
By the definition of equality of ordered pairs, we have and simultaneously solving for a and b
and 
? b = 1
? a = 2
1 B. Question
If (x + 1, 1) = (3y, y - 1), find the values of x and y.
Answer
given: (x + 1, 1) = (3y, y - 1)
To find: values of a and b
By the definition of equality of ordered pairs, we have
x + 1 = 3y and  1 = y - 1
? x = 3y - 1 and  y = 2So,x = 3(2) - 1  = 6 - 1  = 5
? x = 5 and y = 2
2. Question
If the ordered pairs (x, - 1) and (5, y) belong to the set {(a, b): b = 2a - 3}, find the values of x and y.
Answer
given the ordered pairs (x, - 1) and (5, y) belong to the set {(a, b): b = 2a - 3}
To find: values of x and y
solving for first order pair
 (x, - 1) = {(a, b): b = 2a - 3}
 x = a and b = - 1
If b = - 1 then 2a = - 1 + 3 = 2
So, a = 1
 x = 1
Similarly, solving for second order pair
 (5, y) = {(a, b): b = 2a - 3}
 a = 5 and y = b
If a = 5 then b = 2×5 - 3
So, b = 7
 y = 7
3. Question
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.
Answer
given a ? { - 1, 2, 3, 4, 5} and b ? {0, 3, 6},
To find: the ordered pair (a, b) such that a + b = 5
then the ordered pair (a, b) such that a + b = 5 are as follows
(a, b)? {( - 1, 6), (2, 3), (5, 0)}
4. Question
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.
Answer
given that a ? {2, 4, 6, 9} and b ?{4, 6, 18, 27}.
To find: ordered pairs (a, b) such that a divides b and a<b
Here,
2 divides 4, 6, 18 and is also less than all of them
4 divides 4 and is also less than none of them
6 divides 6, 18 and is less than 18 only
9 divides 18, 27 and is less than all of them
Therefore, ordered pairs (a, b) are (2, 4), (2, 6), (2, 18),
(6, 18), (9, 18) and (9, 27)
5. Question
If A = {1, 2} and B = {1, 3}, find A x B and B x A.
Answer
Given A = {1, 2} and B = {1, 3}
To find: A × B, B × A
A × B = {(1, 1), (1, 3), (2, 1), (2, 3)}
B × A = {(1, 1), (1, 2), (3, 1), (3, 2)}
6. Question
Page 3


2. Relations
Exercise 2.1
1 A. Question
If , find the values of a and b.
Answer
Given: 
To find: values of a and b
By the definition of equality of ordered pairs, we have and simultaneously solving for a and b
and 
? b = 1
? a = 2
1 B. Question
If (x + 1, 1) = (3y, y - 1), find the values of x and y.
Answer
given: (x + 1, 1) = (3y, y - 1)
To find: values of a and b
By the definition of equality of ordered pairs, we have
x + 1 = 3y and  1 = y - 1
? x = 3y - 1 and  y = 2So,x = 3(2) - 1  = 6 - 1  = 5
? x = 5 and y = 2
2. Question
If the ordered pairs (x, - 1) and (5, y) belong to the set {(a, b): b = 2a - 3}, find the values of x and y.
Answer
given the ordered pairs (x, - 1) and (5, y) belong to the set {(a, b): b = 2a - 3}
To find: values of x and y
solving for first order pair
 (x, - 1) = {(a, b): b = 2a - 3}
 x = a and b = - 1
If b = - 1 then 2a = - 1 + 3 = 2
So, a = 1
 x = 1
Similarly, solving for second order pair
 (5, y) = {(a, b): b = 2a - 3}
 a = 5 and y = b
If a = 5 then b = 2×5 - 3
So, b = 7
 y = 7
3. Question
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.
Answer
given a ? { - 1, 2, 3, 4, 5} and b ? {0, 3, 6},
To find: the ordered pair (a, b) such that a + b = 5
then the ordered pair (a, b) such that a + b = 5 are as follows
(a, b)? {( - 1, 6), (2, 3), (5, 0)}
4. Question
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.
Answer
given that a ? {2, 4, 6, 9} and b ?{4, 6, 18, 27}.
To find: ordered pairs (a, b) such that a divides b and a<b
Here,
2 divides 4, 6, 18 and is also less than all of them
4 divides 4 and is also less than none of them
6 divides 6, 18 and is less than 18 only
9 divides 18, 27 and is less than all of them
Therefore, ordered pairs (a, b) are (2, 4), (2, 6), (2, 18),
(6, 18), (9, 18) and (9, 27)
5. Question
If A = {1, 2} and B = {1, 3}, find A x B and B x A.
Answer
Given A = {1, 2} and B = {1, 3}
To find: A × B, B × A
A × B = {(1, 1), (1, 3), (2, 1), (2, 3)}
B × A = {(1, 1), (1, 2), (3, 1), (3, 2)}
6. Question
Let A = {1, 2, 3} and B = {3, 4}. Find A x B and show it graphically
Answer
given: A = {1, 2, 3} and B = {3, 4}
To find: graphical representation of A × B
A x B = {(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)}
To represent A × B graphically, given steps must be followed:
a. One horizontal and one vertical axis should be drawn
b. Element of set A should be represented in horizontal axis and on vertical axis elements of set B should be
represented
c. Draw dotted lines perpendicular to horizontal and vertical axes through the elements of set A and B
d. Point of intersection of these perpendicular represents A × B.
7. Question
If A = {1, 2, 3} and B = {2, 4}, what are A x B, B x A, A x A, B x B, and (A x B) n (B x A)?
Answer
given: A = {1, 2, 3} and B = {2, 4}
To find: A × B, B × A, A × A, (A × B) n (B × A)
Now,
A × B = {(1, 2), (1, 4), (2, 2), (2, 4), (3, 2), (3, 4)}
B × A = {(2, 1), (2, 2), (2, 3), (4, 1), (4, 2), (4, 3)}
A × A = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}
B × B = {(2, 2), (2, 4), (4, 2), (4, 4)}
Intersection of two sets represents common elements of both the sets
So,
(A × B) n (B × A) = {(2, 2)}
8. Question
If A and B are two sets having 3 elements in common. If n(A) = 5, n(B) = 4, find n(A x B) and n[(A x B) n (B x
A)].
Answer
given: (A) = 5 and n(B) = 4
To find: [(A × B) n (B×A)]
Page 4


2. Relations
Exercise 2.1
1 A. Question
If , find the values of a and b.
Answer
Given: 
To find: values of a and b
By the definition of equality of ordered pairs, we have and simultaneously solving for a and b
and 
? b = 1
? a = 2
1 B. Question
If (x + 1, 1) = (3y, y - 1), find the values of x and y.
Answer
given: (x + 1, 1) = (3y, y - 1)
To find: values of a and b
By the definition of equality of ordered pairs, we have
x + 1 = 3y and  1 = y - 1
? x = 3y - 1 and  y = 2So,x = 3(2) - 1  = 6 - 1  = 5
? x = 5 and y = 2
2. Question
If the ordered pairs (x, - 1) and (5, y) belong to the set {(a, b): b = 2a - 3}, find the values of x and y.
Answer
given the ordered pairs (x, - 1) and (5, y) belong to the set {(a, b): b = 2a - 3}
To find: values of x and y
solving for first order pair
 (x, - 1) = {(a, b): b = 2a - 3}
 x = a and b = - 1
If b = - 1 then 2a = - 1 + 3 = 2
So, a = 1
 x = 1
Similarly, solving for second order pair
 (5, y) = {(a, b): b = 2a - 3}
 a = 5 and y = b
If a = 5 then b = 2×5 - 3
So, b = 7
 y = 7
3. Question
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.
Answer
given a ? { - 1, 2, 3, 4, 5} and b ? {0, 3, 6},
To find: the ordered pair (a, b) such that a + b = 5
then the ordered pair (a, b) such that a + b = 5 are as follows
(a, b)? {( - 1, 6), (2, 3), (5, 0)}
4. Question
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.
Answer
given that a ? {2, 4, 6, 9} and b ?{4, 6, 18, 27}.
To find: ordered pairs (a, b) such that a divides b and a<b
Here,
2 divides 4, 6, 18 and is also less than all of them
4 divides 4 and is also less than none of them
6 divides 6, 18 and is less than 18 only
9 divides 18, 27 and is less than all of them
Therefore, ordered pairs (a, b) are (2, 4), (2, 6), (2, 18),
(6, 18), (9, 18) and (9, 27)
5. Question
If A = {1, 2} and B = {1, 3}, find A x B and B x A.
Answer
Given A = {1, 2} and B = {1, 3}
To find: A × B, B × A
A × B = {(1, 1), (1, 3), (2, 1), (2, 3)}
B × A = {(1, 1), (1, 2), (3, 1), (3, 2)}
6. Question
Let A = {1, 2, 3} and B = {3, 4}. Find A x B and show it graphically
Answer
given: A = {1, 2, 3} and B = {3, 4}
To find: graphical representation of A × B
A x B = {(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)}
To represent A × B graphically, given steps must be followed:
a. One horizontal and one vertical axis should be drawn
b. Element of set A should be represented in horizontal axis and on vertical axis elements of set B should be
represented
c. Draw dotted lines perpendicular to horizontal and vertical axes through the elements of set A and B
d. Point of intersection of these perpendicular represents A × B.
7. Question
If A = {1, 2, 3} and B = {2, 4}, what are A x B, B x A, A x A, B x B, and (A x B) n (B x A)?
Answer
given: A = {1, 2, 3} and B = {2, 4}
To find: A × B, B × A, A × A, (A × B) n (B × A)
Now,
A × B = {(1, 2), (1, 4), (2, 2), (2, 4), (3, 2), (3, 4)}
B × A = {(2, 1), (2, 2), (2, 3), (4, 1), (4, 2), (4, 3)}
A × A = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}
B × B = {(2, 2), (2, 4), (4, 2), (4, 4)}
Intersection of two sets represents common elements of both the sets
So,
(A × B) n (B × A) = {(2, 2)}
8. Question
If A and B are two sets having 3 elements in common. If n(A) = 5, n(B) = 4, find n(A x B) and n[(A x B) n (B x
A)].
Answer
given: (A) = 5 and n(B) = 4
To find: [(A × B) n (B×A)]
n (A × B) = n(A) × n(B) = 5 x 4 = 20
n (A n B) = 3 (given: A and B has 3 elements in common)
In order to calculate n [(A × B) n (B × A)], we will assume
A = (x, x, x, y, z) and B = (x, x, x, p)
So, we have
(A × B) = {(x, x), (x, x), (x, x), (x, p), (x, x), (x, x), (x, x), (x, p), (x, x), (x, x), (x, x), (x, p), (y, x), (y, x), (y, x),
(y, p), (z, x), (z, x), (z, x), (z, p)}
(B × A) = {(x, x), (x, x), (x, x), (x, y), (x, z), (x, x), (x, x), (x, x), (x, y), (x, z), (x, x), (x, x), (x, x), (x, y), (x, z),
(p, x), (p, x), (p, x), (p, y), (p, z)}
[(A × B) n (B × A)] = {(x, x), (x, x), (x, x), (x, x), (x, x), (x, x), (x, x), (x, x), (x, x)}
? We can say that n [(A × B) n (B × A)] = 9
9. Question
Let A and B be two sets. Show that the sets A x B and B x A have an element in common if the sets A and B
be two sets such that n (A) = 3 and n (B) = 2.
Answer
given: n (A) = 3 n (B) = 2
To prove: The sets A x B and B x A have an element in common if the sets A and B be two sets such that n
(A) = 3 and n (B) = 2
Proof:
Case 1: No elements are common
Assuming:
A = (a, b, c) and B = (e, f)
So, we have:
A × B = {(a, e), (a, f), (b, e), (b, f), (c, e), (c, f)}
B × A = {(e, a), (e, b), (e, c), (f, a), (f, b), (f, c)}
There are no common ordered pair in A × B and B × A.
Case 2: One element is common
Assuming:
A = (a, b, c) and B = (a, f)
So, we have:
A × B = {(a, a), (a, f), (b, a), (b, f), (c, a), (c, f)}
B × A = {(a, a), (a, b), (a, c), (f, a), (f, b), (f, c)}
Here, A × B and B × A have one ordered pair in common.
Therefore, we can say that A × B and B × A will have elements in common if and only if sets A and B have an
element in common.
10. Question
Let A and B be two sets such that n(A) x B, find A and B, where x, y, z are distinct elements
Answer
given: n(A) = 3 and n(B) = 2
To find: distinct elements of set A and B
Page 5


2. Relations
Exercise 2.1
1 A. Question
If , find the values of a and b.
Answer
Given: 
To find: values of a and b
By the definition of equality of ordered pairs, we have and simultaneously solving for a and b
and 
? b = 1
? a = 2
1 B. Question
If (x + 1, 1) = (3y, y - 1), find the values of x and y.
Answer
given: (x + 1, 1) = (3y, y - 1)
To find: values of a and b
By the definition of equality of ordered pairs, we have
x + 1 = 3y and  1 = y - 1
? x = 3y - 1 and  y = 2So,x = 3(2) - 1  = 6 - 1  = 5
? x = 5 and y = 2
2. Question
If the ordered pairs (x, - 1) and (5, y) belong to the set {(a, b): b = 2a - 3}, find the values of x and y.
Answer
given the ordered pairs (x, - 1) and (5, y) belong to the set {(a, b): b = 2a - 3}
To find: values of x and y
solving for first order pair
 (x, - 1) = {(a, b): b = 2a - 3}
 x = a and b = - 1
If b = - 1 then 2a = - 1 + 3 = 2
So, a = 1
 x = 1
Similarly, solving for second order pair
 (5, y) = {(a, b): b = 2a - 3}
 a = 5 and y = b
If a = 5 then b = 2×5 - 3
So, b = 7
 y = 7
3. Question
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.
Answer
given a ? { - 1, 2, 3, 4, 5} and b ? {0, 3, 6},
To find: the ordered pair (a, b) such that a + b = 5
then the ordered pair (a, b) such that a + b = 5 are as follows
(a, b)? {( - 1, 6), (2, 3), (5, 0)}
4. Question
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.
Answer
given that a ? {2, 4, 6, 9} and b ?{4, 6, 18, 27}.
To find: ordered pairs (a, b) such that a divides b and a<b
Here,
2 divides 4, 6, 18 and is also less than all of them
4 divides 4 and is also less than none of them
6 divides 6, 18 and is less than 18 only
9 divides 18, 27 and is less than all of them
Therefore, ordered pairs (a, b) are (2, 4), (2, 6), (2, 18),
(6, 18), (9, 18) and (9, 27)
5. Question
If A = {1, 2} and B = {1, 3}, find A x B and B x A.
Answer
Given A = {1, 2} and B = {1, 3}
To find: A × B, B × A
A × B = {(1, 1), (1, 3), (2, 1), (2, 3)}
B × A = {(1, 1), (1, 2), (3, 1), (3, 2)}
6. Question
Let A = {1, 2, 3} and B = {3, 4}. Find A x B and show it graphically
Answer
given: A = {1, 2, 3} and B = {3, 4}
To find: graphical representation of A × B
A x B = {(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)}
To represent A × B graphically, given steps must be followed:
a. One horizontal and one vertical axis should be drawn
b. Element of set A should be represented in horizontal axis and on vertical axis elements of set B should be
represented
c. Draw dotted lines perpendicular to horizontal and vertical axes through the elements of set A and B
d. Point of intersection of these perpendicular represents A × B.
7. Question
If A = {1, 2, 3} and B = {2, 4}, what are A x B, B x A, A x A, B x B, and (A x B) n (B x A)?
Answer
given: A = {1, 2, 3} and B = {2, 4}
To find: A × B, B × A, A × A, (A × B) n (B × A)
Now,
A × B = {(1, 2), (1, 4), (2, 2), (2, 4), (3, 2), (3, 4)}
B × A = {(2, 1), (2, 2), (2, 3), (4, 1), (4, 2), (4, 3)}
A × A = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}
B × B = {(2, 2), (2, 4), (4, 2), (4, 4)}
Intersection of two sets represents common elements of both the sets
So,
(A × B) n (B × A) = {(2, 2)}
8. Question
If A and B are two sets having 3 elements in common. If n(A) = 5, n(B) = 4, find n(A x B) and n[(A x B) n (B x
A)].
Answer
given: (A) = 5 and n(B) = 4
To find: [(A × B) n (B×A)]
n (A × B) = n(A) × n(B) = 5 x 4 = 20
n (A n B) = 3 (given: A and B has 3 elements in common)
In order to calculate n [(A × B) n (B × A)], we will assume
A = (x, x, x, y, z) and B = (x, x, x, p)
So, we have
(A × B) = {(x, x), (x, x), (x, x), (x, p), (x, x), (x, x), (x, x), (x, p), (x, x), (x, x), (x, x), (x, p), (y, x), (y, x), (y, x),
(y, p), (z, x), (z, x), (z, x), (z, p)}
(B × A) = {(x, x), (x, x), (x, x), (x, y), (x, z), (x, x), (x, x), (x, x), (x, y), (x, z), (x, x), (x, x), (x, x), (x, y), (x, z),
(p, x), (p, x), (p, x), (p, y), (p, z)}
[(A × B) n (B × A)] = {(x, x), (x, x), (x, x), (x, x), (x, x), (x, x), (x, x), (x, x), (x, x)}
? We can say that n [(A × B) n (B × A)] = 9
9. Question
Let A and B be two sets. Show that the sets A x B and B x A have an element in common if the sets A and B
be two sets such that n (A) = 3 and n (B) = 2.
Answer
given: n (A) = 3 n (B) = 2
To prove: The sets A x B and B x A have an element in common if the sets A and B be two sets such that n
(A) = 3 and n (B) = 2
Proof:
Case 1: No elements are common
Assuming:
A = (a, b, c) and B = (e, f)
So, we have:
A × B = {(a, e), (a, f), (b, e), (b, f), (c, e), (c, f)}
B × A = {(e, a), (e, b), (e, c), (f, a), (f, b), (f, c)}
There are no common ordered pair in A × B and B × A.
Case 2: One element is common
Assuming:
A = (a, b, c) and B = (a, f)
So, we have:
A × B = {(a, a), (a, f), (b, a), (b, f), (c, a), (c, f)}
B × A = {(a, a), (a, b), (a, c), (f, a), (f, b), (f, c)}
Here, A × B and B × A have one ordered pair in common.
Therefore, we can say that A × B and B × A will have elements in common if and only if sets A and B have an
element in common.
10. Question
Let A and B be two sets such that n(A) x B, find A and B, where x, y, z are distinct elements
Answer
given: n(A) = 3 and n(B) = 2
To find: distinct elements of set A and B
Also, it is given that {(x, 1), (y, 2), (z, 1)} A × B
Set A has 3 elements whereas Set B has 2 elements.
Also, A × B = {(a, b): a ? A and b ? B}
Therefore, A ? {x, y, z} and B ? {1, 2}
11. Question
Let A = {1, 2, 3, 4} and R = {(a, b): a ? A, b ? A, a divides b}. Write R explicitly.
Answer
given: A = {1, 2, 3, 4} and R = {(a, b): a ? A, b ? A, a divides b}
To find: set R
Both elements of R, a and b, belongs to set A and relation between and elements a and b is that a divides b
So,
1 divides 1, 2, 3 and 4.
2 divides 2 and 4.
3 divides 3.
4 divides 4.
? R = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)}
12. Question
If A = { - 1, 1}, find A x A x A.
Answer
given: A = {-1, 1}
To find: A × A × A
So, A × A = {(-1, -1), (-1, 1), (1, -1), (1, 1)}
And, A × A × A = {(-1, -1, -1), (-1, -1, 1), (-1, 1, -1), (-1, 1, 1), (1, -1, -1), (1, -1, 1), (1, 1, -1), (1, 1,
1)}
13. Question
State whether each of the following statements are true of false. If the statement is false, re - write the given
statement correctly:
i. If P = [m, n} and Q = {n, m} then P x Q = {(m, n), (n, m)}
ii. If A and B are non - empty sets, then A x B is a non - empty set of ordered pairs (x, y) such that x ? B and
y ? A.
iii. If A = {1, 2}, B]{3, 4}, then A x (B n f) = f.
Answer
(i) False
given: P = {m, n} and Q = {n, m}, then
P × Q = {(m, n), (m, m), (n, n), (n, m)}.
(ii) False
given: A and B are non - empty sets
Then A × B is a non - empty set of an ordered pair (x, y) such that x ? A and y ? B
(iii) True