Let `A={1,2,3,5},B={1,2,3}andC={1,2,5}.` Find all the sets X satisfying
`{:((i),X subA","X!inB,(ii),XsubA","X!inC),((iii),XsubB","XneB","X!inC,(iv),XsubA","XsubB","XsubC):}`

Answer

To solve the problem, we need to find all sets \( X \) that satisfy the given conditions based on the sets \( A \), \( B \), and \( C \).

Given:
- \( A = \{1, 2, 3, 5\} \)
- \( B = \{1, 2, 3\} \)
- \( C = \{1, 2, 5\} \)

We will analyze each condition step by step.

### Condition (i): \( X \subseteq A \) and \( X \not\subseteq B \)

1. **Identify elements in \( A \)**: The elements in \( A \) are \( 1, 2, 3, 5 \).
2. **Identify elements in \( B \)**: The elements in \( B \) are \( 1, 2, 3 \).
3. **Elements in \( A \) but not in \( B \)**: The only element in \( A \) that is not in \( B \) is \( 5 \).
4. **Possible subsets of \( A \)**: The subsets of \( A \) are:
- \( \emptyset \)
- \( \{5\} \)
- \( \{1, 5\} \)
- \( \{2, 5\} \)
- \( \{3, 5\} \)
- \( \{1, 2, 5\} \)
- \( \{1, 3, 5\} \)
- \( \{2, 3, 5\} \)
- \( \{1, 2, 3, 5\} \) (this is valid since it contains \( 5 \))

Thus, the valid sets \( X \) satisfying this condition are:
- \( \emptyset \)
- \( \{5\} \)
- \( \{1, 5\} \)
- \( \{2, 5\} \)
- \( \{3, 5\} \)
- \( \{1, 2, 5\} \)
- \( \{1, 3, 5\} \)
- \( \{2, 3, 5\} \)
- \( \{1, 2, 3, 5\} \)

**Total valid sets for condition (i)**: 8 sets.

### Condition (ii): \( X \subseteq A \) and \( X \not\subseteq C \)

1. **Identify elements in \( C \)**: The elements in \( C \) are \( 1, 2, 5 \).
2. **Elements in \( A \) but not in \( C \)**: The element in \( A \) that is not in \( C \) is \( 3 \).
3. **Possible subsets of \( A \)**: We need to include \( 3 \) in our subsets:
- \( \{3\} \)
- \( \{3, 1\} \)
- \( \{3, 2\} \)
- \( \{3, 5\} \)
- \( \{3, 1, 2\} \)
- \( \{3, 1, 5\} \)
- \( \{3, 2, 5\} \)
- \( \{3, 1, 2, 5\} \)
- \( \{1, 2, 3, 5\} \) (this is valid since it contains \( 3 \))

Thus, the valid sets \( X \) satisfying this condition are:
- \( \{3\} \)
- \( \{3, 1\} \)
- \( \{3, 2\} \)
- \( \{3, 5\} \)
- \( \{3, 1, 2\} \)
- \( \{3, 1, 5\} \)
- \( \{3, 2, 5\} \)
- \( \{3, 1, 2, 5\} \)
- \( \{1, 2, 3, 5\} \)

**Total valid sets for condition (ii)**: 8 sets.

### Condition (iii): \( X \subseteq B \) and \( X \not\subseteq C \)

1. **Identify elements in \( B \)**: The elements in \( B \) are \( 1, 2, 3 \).
2. **Elements in \( C \)**: The elements in \( C \) are \( 1, 2, 5 \).
3. **Elements in \( B \) that are not in \( C \)**: The only element in \( B \) that is not in \( C \) is \( 3 \).
4. **Possible subsets of \( B \)**: The subsets of \( B \) that include \( 3 \) are:
- \( \{3\} \)
- \( \{1, 3\} \)
- \( \{2, 3\} \)
- \( \{1, 2, 3\} \)

Thus, the valid sets \( X \) satisfying this condition are:
- \( \{3\} \)
- \( \{1, 3\} \)
- \( \{2, 3\} \)
- \( \{1, 2, 3\} \)

**Total valid sets for condition (iii)**: 4 sets.

### Condition (iv): \( X \subseteq A \), \( X \subseteq B \), and \( X \subseteq C \)

1. **Identify common elements in \( A \), \( B \), and \( C \)**: The common elements are \( 1 \) and \( 2 \).
2. **Possible subsets of \( \{1, 2\} \)**: The subsets are:
- \( \emptyset \)
- \( \{1\} \)
- \( \{2\} \)
- \( \{1, 2\} \)

Thus, the valid sets \( X \) satisfying this condition are:
- \( \emptyset \)
- \( \{1\} \)
- \( \{2\} \)
- \( \{1, 2\} \)

**Total valid sets for condition (iv)**: 4 sets.

### Summary of Results:
- Condition (i): 8 sets
- Condition (ii): 8 sets
- Condition (iii): 4 sets
- Condition (iv): 4 sets