Test: Cartesian Product Of Sets - JEE MCQ

n(A × B) = n(A) × n(B)

n(A) = 3
n(A × A × A) = n(A) * n(A) * n(A)
n(A × A × A) = 3 * 3 * 3 = 27

A = {1, 2} and B = {5, 6, 7} 
A x B = {(1, 5), (1, 6), (1, 7), (2, 5), (2, 6), (2, 7)} 

A = {3, 5}
A × A = {(3, 3), (3, 5), (5, 3), (5, 5)}
A × (A × A) = {(3, 3, 3), (3, 3, 5), (3, 5, 3), (3, 5, 5), (5, 3, 3), (5, 3, 5), (5, 5, 3), (5, 5, 5)}

Cartesian product B x A of two sets A and B is given by

B x A = {(a, b): a ϵ B, b ϵ A}

Hence, B x A = {(5, 3), (5, 4), (6, 3), (6, 4)}

(x – 1, y + 1) = (5, 6)
x – 1 = 5
x = 6
y + 1 = 6
y = 5

A = Φ
B = {0, 1, 2, 3, 4, 5, 6}
A × B = Φ (Cartesian product with an empty set is an empty set)

n(A) = 3
n(B) = 2
(x, 4), (y, 5), (z, 4) are three distinct elements of A x B
So, x, y, z ϵ A
4, 5 ϵ B
A = {x, y, z}
B = {4, 5}

A × B = {(x, a), (x, b), (y, a), (y, b)}
A = {x, y}

A = {2, 4}
B = {5, 6} 
A x B = {(2, 5), (2, 6), (4, 5), (4, 6)}
(A x B) U Φ = A x B = {(2, 5), (2, 6), (4, 5), (4, 6)}

The power set has 2ⁿ elements. For n = 11, size of power set is 2048.

A+2B=7------(1)
3A+2B=9------(2)
Subtracting (2) from (1)
we get, 2A=2
A=1
Subsitute A=1 in (1)
1+2B=7
2B=6
B=3
therefore A = 1, B = 3
 

n(A) = 4
B = {5, 6}
n(B) = 2
n(A x B) = n(A)*n(B) = 4*2 = 8

R × ( P' ∪ Q' )'=R × [( P' )' ∩ ( Q' )' ] =R × (P ∩ Q) = (R × P) ∩ (R × Q)

N = 3
No. of subsets = 2^N = 2³ = 8

It is distributive law.

Correct option is D)

A={1,2,4}, B={2,4,5}, C={2,5}

A−C={1,4}

B−C={4}

(A−C)×(B−C)={1,4}×{4}

={1,4},{4,4}.

To determine if two sets are equal, we need to check if they have exactly the same elements.

Let's analyze each option: a) A = {1, 2} and B = {1} - In this case, set A has two elements: 1 and 2, while set B has only one element: 1. Therefore, the two sets are not equal.

b) A = {1, 2} and B = {1, 2, 3} - Here, set A has two elements: 1 and 2, while set B has three elements: 1, 2, and 3. Thus, the two sets are not equal.

c) A = {1, 2, 3} and B = {2, 1, 3} - In this case, both sets have the same three elements: 1, 2, and 3. The order of the elements does not matter in set theory, so {1, 2, 3} and {2, 1, 3} represent the same set. Therefore, the two sets are equal.

d) A = {1, 2, 4} and B = {1, 2, 3} - In this case, set A has three elements: 1, 2, and 4, while set B has three elements: 1, 2, and 3.

Since the elements 4 and 3 are different, the two sets are not equal.
 

A = {1, 2, 3, 4}, B = {3, 4} and C = {2, 3} 
(A ∩ B) = {3, 4} C = {2, 3}
(A ∩ B x C) = {3,4} x {2,3}
⇒ {(3,2) (3,3) (4,2) (4,3)}