Question

Let N denote the set of natural numbers and A = {n² : n ∈ N} and B = {n³ : n ∈ N}. Which one of the following is not correct?
1. A ∪ B = N
2. The complement of (A ∪ B) is an infinite set
3. A ∩ B must be a finite set
4. A ∩ B must be a proper subset of {m⁶ : m ∈ N}

Answer 1

Correct Answer - Option 1 : A ∪ B = N

Concept:

N denote the set of natural numbers which is an infinite set.

Let A and B be any two sets

A ∪ B = {x: x ∈ A, x ∈ B}

A' = {x: x \(\rm ∈ N , x \notin A\)}

A ∩ B = {x: x \(\rm ∈ A \;\; and \;\; x ∈ B\)}

A proper subset of a set A is a subset of A that is not equal to A

 

Calculations:

Let N denote the set of natural numbers

⇒ N = {1, 2, 3, 4,....}

A = {n2 : n ∈ N}

⇒ A = {1, 4, 9, 16, ....}

B = {n3 : n ∈ N}.

⇒ B = {1, 8, 27, 64, ....}

Consider the statement "A ∪ B = N"

⇒A ∪ B = {1, 4, 8, 9, 16, 27,....} ≠  N

Hence, the statement A ∪ B = N is not true.

 

Consider the statement "The complement of (A ∪ B) is an infinite set"

We know that  The complement of a set, denoted A', is the set of all elements in the given universal set U that are not in A.

⇒ (A ∪ B)' = U - (A ∪ B) = {2, 3, 5, 6,....} = infinite set

⇒ (A ∪ B)' = Infinite set

Hence, the statement "The complement of (A ∪ B) is an infinite set" is true.

 

Consider, the statement "A ∩ B must be a finite set"

A ∩ B  = {1}

Hence, the statement "A ∩ B must be a finite set" is true.

 

Consider, the statement "A ∩ B must be a proper subset of {m6 : m ∈ N}"

Let S = {m6 : m ∈ N}

S = {1, 64, ....}

and A ∩ B  = {1}

A ∩ B must be a proper subset of {m6 : m ∈ N}

Hence, the statement "A ∩ B must be a proper subset of {m6 : m ∈ N}" is True.

Hence,  if N denote the set of natural numbers and A = {n2 : n ∈ N} and B = {n3 : n ∈ N}. then the statement (2), (3) and (4) are correct.