Write the finite and infinite sets from the following sets :
(i) Set of all natural numbers
(ii) Set of all positive integers less than 100
(iii) `{x : x = x^(2)}`
(iv) Set of all lines parallel to x-axis
(v) Set of all prime numbers less then 100
(vi) `{x : x^(2)=16}`
(vii) `{x : x in R,1 le x le 10}`
(viii) Set of all negative integers less than 50.

Answer

To determine whether the given sets are finite or infinite, we will analyze each set one by one.

### Step-by-Step Solution:

1. **Set of all natural numbers**:
- The set of natural numbers is defined as {1, 2, 3, 4, ...}.
- This set continues indefinitely without an upper limit.
- **Conclusion**: This is an **infinite set**.

2. **Set of all positive integers less than 100**:
- This set includes {1, 2, 3, ..., 99}.
- It has a clear upper limit (99) and contains a finite number of elements.
- **Conclusion**: This is a **finite set**.

3. **Set `{x : x = x^2}`**:
- We solve the equation x = x^2.
- Rearranging gives us x^2 - x = 0, which factors to x(x - 1) = 0.
- The solutions are x = 0 and x = 1.
- **Conclusion**: This set contains two elements, so it is a **finite set**.

4. **Set of all lines parallel to the x-axis**:
- Lines parallel to the x-axis can be represented as y = c, where c can be any real number.
- There are infinitely many real numbers, hence infinitely many lines.
- **Conclusion**: This is an **infinite set**.

5. **Set of all prime numbers less than 100**:
- The prime numbers less than 100 are {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}.
- There is a finite count of these primes.
- **Conclusion**: This is a **finite set**.

6. **Set `{x : x^2 = 16}`**:
- Solving x^2 = 16 gives us x = 4 and x = -4.
- There are two solutions.
- **Conclusion**: This is a **finite set**.

7. **Set `{x : x in R, 1 ≤ x ≤ 10}`**:
- This set includes all real numbers between 1 and 10, inclusive.
- There are infinitely many real numbers in this interval.
- **Conclusion**: This is an **infinite set**.

8. **Set of all negative integers less than 50**:
- This set includes {..., -4, -3, -2, -1}.
- There is no upper limit to the negative integers, as they continue indefinitely.
- **Conclusion**: This is an **infinite set**.

### Summary of Results:
- Infinite Sets:
1. Set of all natural numbers
2. Set of all lines parallel to the x-axis
3. Set of all real numbers between 1 and 10
4. Set of all negative integers less than 50

- Finite Sets:
1. Set of all positive integers less than 100
2. Set `{x : x = x^2}`
3. Set of all prime numbers less than 100
4. Set `{x : x^2 = 16}`