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

Correct Answer is : Q⊂PQ PQ⊂P

Correct Answer is : Q⊂PQ PQ⊂P

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COMEDK 2025 Evening Solved Paper

Question : 163 of 180

Marks: +1, -0

P=\{5 m: m \in N\}

Q=\left\{5^m: m \in N\right\}

N

is set of natural numbers, then

Solution:

Concept:

P

is the set of all multiples of

5

, while

Q

is the set of all powers of

5

Explanation:

Write

P = \{5m : m \in \mathbb{N}\} = \{5,10,15,20,25,\dots\}

Write

Q = \{5^{m} : m \in \mathbb{N}\} = \{5,25,125,625,\dots\}

Every element of

Q

is of the form

5^{m} = 5 \cdot 5^{m-1}

, which is clearly a multiple of

5

Thus every element of

Q

belongs to

P

, so

Q \subseteq P

But

P

contains numbers like

10

and

15

which are not powers of

5

, so

P \neq Q

Hence

Q

is a proper subset of

P

, i.e.

Q \subset P

Answer:

Q \subset P

, which corresponds to option C.

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