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Table of Contents – Class 11 Maths Chapter 2 Exercise 2.1 Solutions
Solved Examples before Exercise 2.1
Example 1. If (x + 1, y – 2) = (3,1), find the values of x and y.
Answer 1.
x + 1 = 3, x = 2
y – 2 = 1
y = 3
Example 2. If P = {a, b, c} and Q = {r}, form the sets P × Q and Q × P. Are these two products equal?
Answer 2.
P × Q = {(a, r), (b, r), (c, r)} and Q × P = {(r, a), (r, b), (r, c)}
Since, by the definition of equality of ordered pairs, the pair (a, r) is not equal to the pair (r, a), we conclude that P × Q ≠ Q × P.
Example 3. Let A = {1,2,3}, B = {3,4} and C = {4,5,6}. Find (i) A × (B ∩ C) (ii) (A × B) ∩ (A × C) (iii) A × (B ∪ C) (iv) (A × B) ∪ (A × C)
Answer 3.
(i) By the definition of the intersection of two sets, (B ∩ C) = {4}. Therefore, A × (B ∩ C) = {(1,4), (2,4), (3,4)}.
(ii) Now (A × B) = {(1,3), (1,4), (2,3), (2,4), (3,3), (3,4)} and (A × C) = {(1,4), (1,5), (1,6), (2,4), (2,5), (2,6), (3,4), (3,5), (3,6)} Therefore, (A × B) ∩ (A × C) = {(1, 4), (2, 4), (3, 4)}.
(iii) Since, (B ∪ C) = {3, 4, 5, 6}, we have A × (B ∪ C) = {(1,3), (1,4), (1,5), (1,6), (2,3), (2,4), (2,5), (2,6), (3,3), (3,4), (3,5), (3,6)}.
(iv) Using the sets A × B and A × C from part (ii) above, we obtain (A × B) ∪ (A × C) = {(1,3), (1,4), (1,5), (1,6), (2,3), (2,4), (2,5), (2,6), (3,3), (3,4), (3,5), (3,6)}.
Example 4. If P = {1, 2}, form the set P × P × P.
Answer 4. P× P× P = {(1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,1,2), (2,2,1), (2,2,2)}
Example 5. If R is the set of all real numbers, what do the cartesian products R × R and R × R × R represent?
Answer 5.
The Cartesian product R × R represents the set R × R={(x, y) : x, y ∈ R} which represents the coordinates of all the points in two dimensional space and the cartesian product R × R × R represents the set R × R × R ={(x, y, z) : x, y, z ∈ R} which represents the coordinates of all the points in three-dimensional space.
Example 6. If A × B ={(p, q),(p, r), (m, q), (m, r)}, find A and B
Answer 6. A = set of first elements = {p, m}
B = set of second elements = {q, r}
Exercise 2.1 solved questions
Question 1. If (x/3 + 1, y- 2/3) = (5/3, 1/3), find the values of x and y
Answer 1.
x/3 + 1 = 5/3
x/3 = 5/3 – 1 = 2/3
x = 2
y – 2/3 = 1/3
y = 1
Question 2. If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of elements in (A×B)
Answer 2. n(A) = 3
n(B) = 3
n(A×B)=3 × 3 = 9
Question 3. If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G
Answer 3.
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) }
Question 4. State whether each of the following statements are 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
Answer 4.
(i) False, P × Q = {(m, n),(m, m)}, (n,n), (n,m)}
(ii) True
(iii) True
Question 5. If A = {–1, 1}, find A × A × A.
Answer 5.
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) }
Question 6. If A × B = {(a, x),(a , y), (b, x), (b, y)}. Find A and B
Answer 6.
A = {a,b}
B = {x,y}
Question 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.
Answer 7.
(i) B ∩ C = φ
A × φ = φ
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)}
(A × B) ∩ (A × C) = φ
(ii) A × C = {(1,5), (1,6), (2,5), (2,6)}
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)}
A × C is a subset of B × D
Question 8.
Let A = {1, 2} and B = {3, 4}. Write A × B. How many subsets will A × B have? List them
Answer 8. A × B = {(1,3), (1,4), (2,3), (2,4)}
16 subsets
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)}
Question 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.
Answer 9.
A = {x,y,z}
B = {1,2}
Question 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.
Answer 10.
n(A × A) = 3 × 3 = 9
A = {-1,0,1}
A × A = {−1, 0, 1} × {−1, 0, 1}
= { (−1, −1), (−1, 0), (−1, 1),
(0, −1), (0, 0), (0, 1),
(1, −1), (1, 0), (1, 1) }
∴ The remaining elements of A × A other than the given two elements
(−1, 0) and (0, 1) of A × A are
(−1, −1), (−1, 1), (0, −1), (0, 0),
(1, −1), (1, 0), (1, 1).