From the given sets, which is an infinite set:

1. $\{x:x\in N~and~(x-1)(x-2)=0\}$

2. $\{x: x \in N ~ and ~ x ~ is ~prime ~number~ and ~less ~than ~199\}$

3. $\{x:x\in N ~and~ x^{5}-1=0\}$ 

4. $\{x:x\in N~and~x~is~odd\}$

Solution:

We check each option one by one: 1. \(\{x \in \mathbb{N} : (x-1)(x-2)=0\} = \{1,2\}\), which is a finite set.

2. \(\{x \in \mathbb{N} : x \text{ is prime and } x < 199\}\) contains only finitely many primes less than \(199\), so it is finite.

3. \(\{x \in \mathbb{N} : x^{5}-1=0\} \;\Rightarrow\; x^{5}=1 \;\Rightarrow\; x=1\). Thus the set is \(\{1\}\), which is finite.

4. \(\{x \in \mathbb{N} : x \text{ is odd}\} = \{1,3,5,7,\dots\}\), which is an infinite set.

Answer: Option (4).