Question

If \( P = \{5m : m \in N\} \) and \( Q = \{5^m : m \in N\} \), where \( N \) is set of natural numbers, then

• A. \( P = Q \)
• B. \( P \subset Q \)
• C. \( Q \subset P \)
• D. \( P \cup Q = N \)

Hint:

Multiples of a number form a larger set than its powersAlways check with a simple counterexample to verify subset relations.

Answer 1

The Correct Option is C

Step 1: Write elements of set \( P \).
\[ P = \{5m : m \in N\} \]
So elements are:
\[ 5, 10, 15, 20, 25, \ldots \]

Step 2: Write elements of set \( Q \).
\[ Q = \{5^m : m \in N\} \]
So elements are:
\[ 5, 25, 125, 625, \ldots \]

Step 3: Compare both sets.
Every element of \( Q \) is a multiple of 5.
So all elements of \( Q \) belong to \( P \).

Step 4: Check reverse inclusion.
Not all multiples of 5 are powers of 5.
For example:
\[ 10 \in P \text{ but } 10 \notin Q \]

Step 5: Conclusion about sets.
\[ Q \subset P \]

Step 6: Match with options.
Correct option is (C).

Step 7: Final conclusion.
\[ \boxed{Q \subset P} \]