From the given sets, which is an infinite set:$$\begin{aligned}\text{(a)} & \quad \{x : x \in \mathbb{N} \text{ and } (x-1)(x-2)=0 \} \\ \text{(b)} & \quad \{x : x \in \mathbb{N} \text{ and $x$ is a prime number with } x < 199 \} \\ \text{(c)} & \quad \{x : x \in \mathbb{N} \text{ and } x^{5} - 1 = 0 \} \\ \text{(d)} & \quad \{x : x \in \mathbb{N} \text{ and $x$ is odd} \}\end{aligned}$$
Explanation:
We need to check each option:
(a) \(\{x : x \in \mathbb{N} \text{ and } (x-1)(x-2)=0 \}\)
⇒ Solutions: \(x=1\) or \(x=2\).
So, this set = \(\{1,2\}\), which is finite.
(b) \(\{x : x \in \mathbb{N} \text{ and $x$ is a prime number with } x < 199 \}\)
There are finitely many primes less than 199.
So, this set is finite.
(c) \(\{x : x \in \mathbb{N} \text{ and } x^{5}-1=0 \}\)
Solving: \(x^{5} = 1 \Rightarrow x=1\).
So, this set = \(\{1\}\), which is finite.
(d) \(\{x : x \in \mathbb{N} \text{ and $x$ is odd} \}\)
Odd natural numbers are \(1,3,5,7, \dots\).
This set goes on infinitely.
So, this set is infinite.
\[ \therefore \;\; \text{Correct answer is (d).} \]
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