NCERT Solutions for Class 11 Maths Chapter 2 Relations And Functions Exercise 2.1
Below is the NCERT Solutions for Class 11 Maths Chapter 2 Relations And Functions Exercise 2.1:
1. If
, find the values of
x
and
y
.
Solution:
Given,
As the ordered pairs are equal, the corresponding elements should also be equal.
Thus,
x/3 + 1 = 5/3 and y – 2/3 = 1/3
Solving, we get
x + 3 = 5 and 3y – 2 = 1 [Taking L.C.M and adding]
x = 2 and 3y = 3
Therefore,
x = 2 and y = 1
2. If set A has 3 elements and set B = {3, 4, 5}, then find the number of elements in (A × B).
Solution:
Given, set A has 3 elements, and the elements of set B are {3, 4, and 5}.
So, the number of elements in set B = 3
Then, the number of elements in (A × B) = (Number of elements in A) × (Number of elements in B)
= 3 × 3 = 9
Therefore, the number of elements in (A × B) will be 9.
3. If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G.
Solution:
Given, G = {7, 8} and H = {5, 4, 2}
We know that
The Cartesian product of two non-empty sets, P and Q, is given as
P × Q = {(
p
,
q
):
p
∈ P,
q
∈ Q}
So,
G × H = {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)}
H × G = {(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)}
4. State whether each of the following statements is true or false. If the statement is false, rewrite the given statement correctly.
(i) If P = {
m
,
n
} and Q = {
n
,
m
}, then P × Q = {(
m
,
n
), (
n
,
m
)}.
(ii) If A and B are non-empty sets, then A × B is a non-empty set of ordered pairs (
x
,
y
) such that
x
∈ A and
y
∈ B.
(iii) If A = {1, 2}, B = {3, 4}, then A × (B ∩ Φ) = Φ.
Solution:
(i) The statement is False. The correct statement is
If P = {
m
,
n
} and Q = {
n
,
m
}, then
P × Q = {(
m
,
m
), (
m
,
n
), (
n,
m
), (
n
,
n
)}
(ii) True
(iii) True
5. If A = {–1, 1}, find A × A × A.
Solution:
The A × A × A for a non-empty set A is given by
A × A × A = {(
a
,
b
,
c
):
a
,
b
,
c
∈ A}
Here, it is given A = {–1, 1}
So,
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)}
6. If A × B = {(
a
,
x
), (
a
,
y
), (
b
,
x
), (
b
,
y
)}. Find A and B.
Solution:
Given,
A × B = {(
a
,
x
), (
a,
y
), (
b
,
x
), (
b
,
y
)}
We know that the Cartesian product of two non-empty sets, P and Q, is given by
P × Q = {(
p
,
q
):
p
∈ P,
q
∈ Q}
Hence, A is the set of all first elements, and B is the set of all second elements.
Therefore, A = {
a
,
b
} and B = {
x
,
y
}
7. Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that
(i) A × (B ∩ C) = (A × B) ∩ (A × C)
(ii) A × C is a subset of B × D
Solution:
Given,
A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}
(i) To verify: A × (B ∩ C) = (A × B) ∩ (A × C)
Now, B ∩ C = {1, 2, 3, 4} ∩ {5, 6} = Φ
Thus,
L.H.S. = A × (B ∩ C) = A × Φ = Φ
Next,
A × B = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)}
A × C = {(1, 5), (1, 6), (2, 5), (2, 6)}
Thus,
R.H.S. = (A × B) ∩ (A × C) = Φ
Therefore, L.H.S. = R.H.S.
Hence, verified
(ii) To verify: A × C is a subset of B × D
First,
A × C = {(1, 5), (1, 6), (2, 5), (2, 6)}
And,
B × D = {(1, 5), (1, 6), (1, 7), (1, 8), (2, 5), (2, 6), (2, 7), (2, 8), (3, 5), (3, 6), (3, 7), (3, 8), (4, 5), (4, 6), (4, 7), (4, 8)}
Now, it’s clearly seen that all the elements of set A × C are the elements of set B × D.
Thus, A × C is a subset of B × D.
Hence, verified
8. Let A = {1, 2} and B = {3, 4}. Write A × B. How many subsets will A × B have? List them.
Solution:
Given,
A = {1, 2} and B = {3, 4}
So,
A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}
Number of elements in A × B is
n
(A × B) = 4
We know that
If C is a set with
n
(C) =
m
, then
n
[P(C)] = 2
m
.
Thus, the set A × B has 2
4
= 16 subsets
And these subsets are as below:
Φ, {(1, 3)}, {(1, 4)}, {(2, 3)}, {(2, 4)}, {(1, 3), (1, 4)}, {(1, 3), (2, 3)}, {(1, 3), (2, 4)}, {(1, 4), (2, 3)}, {(1, 4), (2, 4)}, {(2, 3), (2, 4)}, {(1, 3), (1, 4), (2, 3)}, {(1, 3), (1, 4), (2, 4)}, {(1, 3), (2, 3), (2, 4)}, {(1, 4), (2, 3), (2, 4)}, {(1, 3), (1, 4), (2, 3), (2, 4)}
9. 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
and
z
are distinct elements.
Solution:
Given,
n
(A) = 3 and
n
(B) = 2; and (
x
, 1), (
y
, 2), (
z
, 1) are in A × B.
We know that
A = Set of first elements of the ordered pair elements of A × B
B = Set of second elements of the ordered pair elements of A × B.
So, clearly,
x
,
y
, and
z
are the elements of A; and
1 and 2 are the elements of B.
As
n
(A) = 3 and
n
(B) = 2, it is clear that set A = {
x
,
y
,
z
} and set B = {1, 2}.
10. The Cartesian product A × A has 9 elements, among which are found (–1, 0) and (0, 1). Find the set A and the remaining elements of A × A.
Solution:
We know that
If
n
(A) =
p
and
n
(B) =
q,
then
n
(A × B) =
pq
.
Also,
n
(A × A) =
n
(A) ×
n
(A)
Given,
n
(A × A) = 9
So,
n
(A) ×
n
(A) = 9
Thus,
n
(A) = 3
Also, given that the ordered pairs (–1, 0) and (0, 1) are two of the nine elements of A × A.
And, we know in A × A = {(
a, a
):
a
∈ A}.
Thus, –1, 0, and 1 have to be the elements of A.
As
n
(A) = 3, clearly A = {–1, 0, 1}
Hence, the remaining elements of set A × A are as follows:
(–1, –1), (–1, 1), (0, –1), (0, 0), (1, –1), (1, 0), and (1, 1)
Benefits of Solving NCERT Solutions for Class 11 Maths Chapter 2 Exercise 2.1
-
Clear Understanding of Concepts:
The exercise focuses on Relations and Functions, and solving these problems helps students develop a deep understanding of important mathematical concepts like domain, codomain, and range.
-
Improved Problem-Solving Skills
: By practicing the problems, students learn how to apply theoretical knowledge to practical problems, enhancing their problem-solving abilities.
-
Stronger Foundation
: This exercise builds a solid foundation for advanced topics in mathematics, especially those related to functions, which are crucial for further studies in calculus, algebra, and other branches of mathematics.
-
Boosts Exam Preparation
: Regularly solving exercises from NCERT helps students familiarize themselves with the types of questions commonly asked in exams, improving accuracy and speed during the actual exam.
-
Step-by-Step Solutions
: The NCERT solutions provide detailed, step-by-step explanations, making it easier for students to understand how to approach each problem and ensuring they don’t miss any important steps.
-
Enhances Confidence
: As students solve more problems correctly, their confidence in tackling related topics grows, which positively impacts their overall learning and performance in mathematics.
