Cartesian Product of Sets MCQ Quiz in বাংলা - Objective Question with Answer for Cartesian Product of Sets - বিনামূল্যে ডাউনলোড করুন [PDF]
Last updated on Mar 15, 2025
Latest Cartesian Product of Sets MCQ Objective Questions
Top Cartesian Product of Sets MCQ Objective Questions
Cartesian Product of Sets Question 1:
If A = [2, 5] and B = [3, 8], then what is the value of A × B?
Answer (Detailed Solution Below)
Cartesian Product of Sets Question 1 Detailed Solution
Given:
A = [2, 5] and B = [3, 8]
Calculation:
A × B = [2, 5] × [3, 8]
A × B = [(2, 3), (2, 8), (5, 3), (5, 8)]
∴ Correct option is 2.
Cartesian Product of Sets Question 2:
Let A = {1, 3, 5}, B = {4, 6} and C = {5, 6, 7}. Find A × (B ∩ C)
Answer (Detailed Solution Below)
Cartesian Product of Sets Question 2 Detailed Solution
Concept:
Intersection of sets:
Let A and B be two sets. The intersection of A and B is the set of all those elements which are present in both sets A and B.
The intersection of A and B is denoted by A ∩ B
i.e A ∩ B = {x : x ∈ A and x ∈ B}
The Venn diagram for intersection is as shown below:
Cartesian Product: The Cartesian product of two sets A and B, denoted A × B, is the set of all possible ordered pairs where the elements of A are first and the elements of B are second.
In set-builder notation, A × B = {(a, b) : a ∈ A and b ∈ B}.
Calculation:
Given: A = {1, 3, 5}, B = {4, 6} and C = {5, 6, 7}
B ∩ C = {6}
A × (B ∩ C) = {1, 3, 5} × {6}
⇒ {(1, 6) (3, 6) (5, 6)}
Cartesian Product of Sets Question 3:
X × Y = { (a, b); a belongs to X, b belongs to Y} IXI = n, IYI = m, then What is the value of IX × YI = ?
Answer (Detailed Solution Below)
Cartesian Product of Sets Question 3 Detailed Solution
Concept use:
"X × Y = { (a, b); a belongs to X, b belongs to Y}" is the concept of a Cartesian product of two sets.
The Cartesian product of two sets X and Y is the set of all possible ordered pairs where the first element is from set X and the second element is from set Y.
The notation |X| denotes the cardinality of a set X, which means the number of elements in the set. Similarly, |Y| denotes the number of elements in set Y.
Calculations:
|X| = n, which means there are n elements in set X. |Y| = m, means there are m elements in set Y.
The cardinality of the Cartesian product of two sets |X × Y| is equal to the product of the cardinalities of each individual set. This is because for each element of set X, we can form a pair with each element of set Y.
So if there are n elements in set X and m elements in set Y, the total number of possible pairs in the Cartesian product X × Y would be n × m, namely every element of X paired with every element of Y.
Hence, the value of |X × Y| = n × m = mn
Cartesian Product of Sets Question 4:
If A = [2, 5] and B = [3, 8], then what is the value of A × B?
Answer (Detailed Solution Below)
Cartesian Product of Sets Question 4 Detailed Solution
Given:
A = [2, 5] and B = [3, 8]
Calculation:
A × B = [2, 5] × [3, 8]
A × B = [(2, 3), (2, 8), (5, 3), (5, 8)]
∴ Correct option is 2.
Cartesian Product of Sets Question 5:
If (x2 - 3x + 5, y – 4) = (3,1), find the values of x and y.
Answer (Detailed Solution Below)
Cartesian Product of Sets Question 5 Detailed Solution
Concept:
The ordered pairs are equal, the corresponding elements are equal.
(a, b) = (c, d)
Then a = c and b = d
Calculation:
Given: (x2 - 3x + 5, y – 4) = (3,1)
The ordered pairs are equal, the corresponding elements are equal.
x2 - 3x + 5 = 3
⇒ x2 - 3x + 2 = 0
⇒ x2 - 2x - x + 2 = 0
⇒ x(x - 2) -1(x - 2) = 0
⇒ (x - 1)(x - 2) = 0
⇒ x = 1, 2
And y - 4 = 1
⇒ y = 5
Cartesian Product of Sets Question 6:
If B = [-1, 5] then find of B × B × B is
Answer (Detailed Solution Below)
Cartesian Product of Sets Question 6 Detailed Solution
Concept:
For any two non-empty sets A and B, we have:
- A X B = {(a, b) | a ∈ A and b ∈ B}
- B X A = {(b, a) | a ∈ A and b ∈ B}
Calculation:
Given: B = [-1, 5]
B × B = [-1, 5] × [-1, 5] = [(-1, -1), (-1, 5), (5, -1), (5, 5)]
B × B × B= [-1, 5] × [-1, 5] × [-1, 5] = [(-1, -1), (-1, 5), (5, -1), (5, 5)] × [-1, 5]
= [(-1, -1, -1), (-1, -1, 5), (-1, 5, -1), (-1, 5, 5), (5, -1, -1), (5, -1, 5), (5, 5, -1), (5, 5, 5)]
Cartesian Product of Sets Question 7:
Let A = {1, 3, 5}, B = {4, 5} and C = {4, 5, 6}. Find A × (B ∪ C).
Answer (Detailed Solution Below)
Cartesian Product of Sets Question 7 Detailed Solution
Concept:
Union of sets: Let A and B be two sets.
The union of A and B is the set of all those elements which belong to either A or B or both A and B.
The union of A and B is denoted by A ∪ B.
i.e A ∪ B = {x : x ∈ A or x ∈ B}
The Venn diagram for the union of any two sets is shown below:
Cartesian Product: The Cartesian product of two sets A and B, denoted A × B, is the set of all possible ordered pairs where the elements of A are first and the elements of B are second.
In set-builder notation, A × B = {(a, b) : a ∈ A and b ∈ B}.
Calculation:
Given: A = {1, 3, 5}, B = {4, 5} and C = {4, 5, 6}
B U C = {4, 5, 6}
A × (B U C) = {1, 3, 5} × {4, 5, 6}
⇒ {(1, 4) (1, 5) (1, 6) (3, 4) (3, 5) (3, 6) (5, 4) (5, 5) (5, 6)}
Cartesian Product of Sets Question 8:
The figure shows a relationship between the sets P and Q. Write this relation in roster form?
Answer (Detailed Solution Below)
Cartesian Product of Sets Question 8 Detailed Solution
Concept:
Tabular form / Roaster form:
In this method, a set is described by listing all the elements, separated by commas, within the braces {}.
Example: A = {2, 3, 5} is a set of first three prime numbers.
Set - Builder form:
In this method, all the elements of the set possess a single common property, which is being enlisted.
Example: B = {x : 6 ≤ x ∈ N ≤ 12}
Calculation:
Given:
R = {(8, 4), (9, 5), (10, 6), (11, 7)}
Cartesian Product of Sets Question 9:
The figure shows a relationship between the sets P and Q. Write this relation in set - builder form?
Answer (Detailed Solution Below)
Cartesian Product of Sets Question 9 Detailed Solution
Concept:
Tabular form / Roaster form:
In this method, a set is described by listing all the elements, separated by commas, within the braces {}.
Example: A = {2, 3, 5} is a set of first three prime numbers.
Set - Builder form:
In this method, all the elements of the set possess a single common property, which is being enlisted.
Example: B = {x : 6 ≤ x ∈ N ≤ 12}
Calculation:
Given:
As we can see that, all the elements of Q are subtract of P
R = {(P, Q) : Q = P – 4 for P = 8, 9, 10, 11}
Cartesian Product of Sets Question 10:
If \(\rm \left({\frac {x}{4} \ + 1, y \ - \frac{1}{4}}\right) = \left(\frac {5}{4}, \frac {3}{4}\right)\), find the values of x and y
Answer (Detailed Solution Below)
Cartesian Product of Sets Question 10 Detailed Solution
Concept:
The ordered pairs are equal, the corresponding elements are equal.
(a, b) = (c, d)
Then a = c and b = d
Calculation:
Given: \(\rm \left({\frac {x}{4} \ + 1, y \ - \frac{1}{4}}\right) = \left(\frac {5}{4}, \frac {3}{4}\right)\)
The ordered pairs are equal, the corresponding elements are equal.
= \(\rm \frac {x}{4} + 1 = \frac {5}{4}\)
= \(\rm \frac {x\ +\ 4}{4} = \frac {5}{4}\)
= x = 5 - 4 = 1
= x = 1
And, \(\rm y \ - \ \frac {1}{4} = \frac {3}{4}\)
= \(\rm \frac {4y \ - \ 1}{4} = \frac {3}{4}\)
= 4y - 1 = 3
= 4y = 4
= y = 1