Question

If \(P = \{3m : m \in \mathbb{N}\}\) and  \(Q = \{3m : m \in \mathbb{N}\}\) are two sets, then

• A. \(P ⊂ Q\)
• B. P = Q
• C. \(Q ⊂P\)
• D. \(P \cup Q = \mathbb{N}\)

Answer 1

The Correct Option is B

We are given two sets: 

\( P = \{3m : m \in \mathbb{N}\} \) and \( Q = \{3m : m \in \mathbb{N}\} \).

Step 1: Understanding the Sets

Both sets are defined using the same condition: all elements are multiples of 3, where \( m \) is a natural number.

Since both sets are identical, we conclude that: \[ P = Q \]

Step 2: Evaluating the Options

• Option 1: \( P \subset Q \) (True, but since \( P = Q \), it is not the best choice)
• Option 2: \( Q \subset P \) (Same reasoning as above)
• Option 3: \( P = Q \) (Correct, since both sets are identical)
• Option 4: \( P \cup Q = \mathbb{N} \) (Incorrect, because \( P \) and \( Q \) contain only multiples of 3, not all natural numbers)

Final Answer: \( P = Q \).

Answer 2

To determine the relationship between the sets \(P\) and \(Q\), we start by examining their definitions:

• Set \(P\) is defined as \(P = \{3m : m \in \mathbb{N}\}\). This means \(P\) consists of all positive multiples of 3.
• Set \(Q\) is defined as \(Q = \{3m : m \in \mathbb{N}\}\). Similarly, \(Q\) consists of all positive multiples of 3.

Since both sets \(P\) and \(Q\) are defined identically, we conclude that \(P = Q\). 
They contain the same elements, which are all positive multiples of 3.

Thus, the correct answer is: P = Q