If x = {1,2,3 …., 10 } and a represents any elements of X then write the follwing sets containing all the elements satisfing the given conditions
` a in X but a^2 in X `
`a in X but a//2 in X `
a is factor of 24

Explanation

To solve the given problem, we need to find the sets based on the conditions provided. Let's break it down step by step.

### Given:
- \( X = \{1, 2, 3, \ldots, 10\} \)
- \( a \) represents any element of \( X \)

### Part 1: Find the set where \( a \in X \) but \( a^2 \in X \)

1. **Identify the elements of \( X \)**:
- The elements of \( X \) are \( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \).

2. **Check each element \( a \) in \( X \)**:
- For \( a = 1 \):
- \( a^2 = 1^2 = 1 \) (which is in \( X \))
- For \( a = 2 \):
- \( a^2 = 2^2 = 4 \) (which is in \( X \))
- For \( a = 3 \):
- \( a^2 = 3^2 = 9 \) (which is in \( X \))
- For \( a = 4 \):
- \( a^2 = 4^2 = 16 \) (which is not in \( X \))
- For \( a = 5 \):
- \( a^2 = 5^2 = 25 \) (which is not in \( X \))
- For \( a = 6 \):
- \( a^2 = 6^2 = 36 \) (which is not in \( X \))
- For \( a = 7 \):
- \( a^2 = 7^2 = 49 \) (which is not in \( X \))
- For \( a = 8 \):
- \( a^2 = 8^2 = 64 \) (which is not in \( X \))
- For \( a = 9 \):
- \( a^2 = 9^2 = 81 \) (which is not in \( X \))
- For \( a = 10 \):
- \( a^2 = 10^2 = 100 \) (which is not in \( X \))

3. **Collect valid elements**:
- The valid elements where \( a^2 \) is also in \( X \) are \( 1, 2, 3 \).

4. **Final set for Part 1**:
- \( A = \{1, 2, 3\} \)

### Part 2: Find the set where \( a \in X \) but \( \frac{a}{2} \in X \)

1. **Check each element \( a \) in \( X \)**:
- For \( a = 1 \):
- \( \frac{1}{2} \) (not in \( X \))
- For \( a = 2 \):
- \( \frac{2}{2} = 1 \) (which is in \( X \))
- For \( a = 3 \):
- \( \frac{3}{2} = 1.5 \) (not in \( X \))
- For \( a = 4 \):
- \( \frac{4}{2} = 2 \) (which is in \( X \))
- For \( a = 5 \):
- \( \frac{5}{2} = 2.5 \) (not in \( X \))
- For \( a = 6 \):
- \( \frac{6}{2} = 3 \) (which is in \( X \))
- For \( a = 7 \):
- \( \frac{7}{2} = 3.5 \) (not in \( X \))
- For \( a = 8 \):
- \( \frac{8}{2} = 4 \) (which is in \( X \))
- For \( a = 9 \):
- \( \frac{9}{2} = 4.5 \) (not in \( X \))
- For \( a = 10 \):
- \( \frac{10}{2} = 5 \) (which is in \( X \))

2. **Collect valid elements**:
- The valid elements where \( \frac{a}{2} \) is also in \( X \) are \( 2, 4, 6, 8, 10 \).

3. **Final set for Part 2**:
- \( B = \{2, 4, 6, 8, 10\} \)

### Part 3: Find the set where \( a \) is a factor of 24

1. **Identify the factors of 24**:
- The factors of 24 are \( 1, 2, 3, 4, 6, 8, 12, 24 \).

2. **Select factors that are in \( X \)**:
- From the factors, the elements that are also in \( X \) are \( 1, 2, 3, 4, 6, 8 \).

3. **Final set for Part 3**:
- \( C = \{1, 2, 3, 4, 6, 8\} \)

### Summary of Sets:
- \( A = \{1, 2, 3\} \)
- \( B = \{2, 4, 6, 8, 10\} \)
- \( C = \{1, 2, 3, 4, 6, 8\} \)