Which of the following is infinite set?

Question

Accepted Answer

The correct answer is Option 4: $\{x:x\in N; x$ is odd$\}$.

Explanation:

1. **Option 4: $\{x:x\in N; x$ is odd$\}$**
   - This set includes all natural numbers that are odd. The sequence of odd natural numbers is 1, 3, 5, 7, 9, and so on. There is no upper limit to the natural numbers, and thus, no upper limit to the odd natural numbers. Therefore, this set is infinite because there are infinitely many odd numbers in the set of natural numbers.

2. **Option 1: $\{x:x\in N;(x-1)(x-2)=0\}$**
   - This set is defined by the equation $(x-1)(x-2)=0$, which implies $x-1=0$ or $x-2=0$. Solving these equations gives $x=1$ or $x=2$. Therefore, the set is $\{1, 2\}$, which is a finite set containing only two elements.

3. **Option 2: $\{x:x\in N;x^{2}=4\}$**
   - This set is defined by the equation $x^2=4$. Solving this equation gives $x=2$ (since $x$ must be a natural number). Therefore, the set is $\{2\}$, which is a finite set containing only one element.

4. **Option 3: $\{x:x\in N;2x-1=0\}$**
   - This set is defined by the equation $2x-1=0$. Solving this equation gives $x=\frac{1}{2}$. However, since $x$ must be a natural number, there are no natural numbers that satisfy this equation. Therefore, this set is empty, which is a finite set with zero elements.

In conclusion, Option 4 is the only option that describes an infinite set, as it includes all odd natural numbers, which are infinite in quantity. The other options describe finite sets with a limited number of elements.