Q105 / 180 · COMEDK 2025 (10 May Shift 3) · Mathematics
If $P=\{5 m: m \in N\}$ and $Q=\left\{5^m: m \in N\right\}$, where $N$ is set of natural numbers, then
- Correct answerA$Q \subset P$
- B$P=Q$
- C$P \subset Q$
- D$P \cup Q=N$
Solution
The set $P$ is defined as $P = \{5m : m \in N\} = \{5, 10, 15, 20, 25, 30, \dots\}$.
The set $Q$ is defined as $Q = \{5^m : m \in N\} = \{5^1, 5^2, 5^3, \dots\} = \{5, 25, 125, 625, \dots\}$.
Comparing the elements of $P$ and $Q$, we observe that every element of $Q$ is of the form $5^m$. Since $m \in N$, $5^m = 5 \times 5^{m-1}$. Let $k = 5^{m-1}$. Since $m \ge 1$, $k$ is a natural number. Thus, $5^m = 5k$, which is an element of $P$.
Therefore, every element of $Q$ is also an element of $P$, which implies $Q \subset P$.
Checking other options, $P$ contains elements like $10$ which are not in $Q$, so $P \neq Q$ and $P \not\subset Q$. Also, $P \cup Q$ contains only multiples of $5$ and powers of $5$, which is a proper subset of $N$.
Answer: $Q \subset P$
The set $Q$ is defined as $Q = \{5^m : m \in N\} = \{5^1, 5^2, 5^3, \dots\} = \{5, 25, 125, 625, \dots\}$.
Comparing the elements of $P$ and $Q$, we observe that every element of $Q$ is of the form $5^m$. Since $m \in N$, $5^m = 5 \times 5^{m-1}$. Let $k = 5^{m-1}$. Since $m \ge 1$, $k$ is a natural number. Thus, $5^m = 5k$, which is an element of $P$.
Therefore, every element of $Q$ is also an element of $P$, which implies $Q \subset P$.
Checking other options, $P$ contains elements like $10$ which are not in $Q$, so $P \neq Q$ and $P \not\subset Q$. Also, $P \cup Q$ contains only multiples of $5$ and powers of $5$, which is a proper subset of $N$.
Answer: $Q \subset P$
Hence, the correct answer is option: 'A'.