Cartesian Product of Sets MCQ Quiz in বাংলা - Objective Question with Answer for Cartesian Product of Sets - বিনামূল্যে ডাউনলোড করুন [PDF]

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.

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)}

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

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.

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

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)]

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)}

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)}

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}

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