If `A = {1,2,3,4},B={2,3,4} and C={1,3}` , then find the set X such that :
(i) `X subB and X sub C`
(ii) `X sub B and X cancel(sub)C`
(iii)`X subA, X subB and X sub C`.

Answer

To solve the problem step by step, we will analyze each condition given for the set \( X \) based on the sets \( A \), \( B \), and \( C \).

### Given Sets:
- \( A = \{1, 2, 3, 4\} \)
- \( B = \{2, 3, 4\} \)
- \( C = \{1, 3\} \)

### (i) Find \( X \) such that \( X \subseteq B \) and \( X \subseteq C \)

1. **Identify the elements in \( B \) and \( C \)**:
- \( B = \{2, 3, 4\} \)
- \( C = \{1, 3\} \)

2. **Find the common elements in \( B \) and \( C \)**:
- The common elements are those that are present in both sets.
- The only common element is \( 3 \).

3. **Include the empty set**:
- The empty set \( \emptyset \) is a subset of every set.

4. **Conclusion for (i)**:
- Thus, \( X = \{3\} \) and \( X = \emptyset \).

### (ii) Find \( X \) such that \( X \subseteq B \) and \( X \cancel{\subseteq} C \)

1. **Identify elements in \( B \) that are not in \( C \)**:
- From \( B = \{2, 3, 4\} \) and \( C = \{1, 3\} \), we see that:
- \( 2 \) is in \( B \) but not in \( C \).
- \( 3 \) is in both \( B \) and \( C \) (not suitable).
- \( 4 \) is in \( B \) but not in \( C \).

2. **Possible subsets of \( B \) excluding elements in \( C \)**:
- The valid elements are \( 2 \) and \( 4 \).
- The possible subsets of \( \{2, 4\} \) are:
- \( \emptyset \)
- \( \{2\} \)
- \( \{4\} \)
- \( \{2, 4\} \)

3. **Conclusion for (ii)**:
- Thus, \( X = \emptyset, \{2\}, \{4\}, \{2, 4\} \).

### (iii) Find \( X \) such that \( X \subseteq A \), \( X \subseteq B \), and \( X \subseteq C \)

1. **Identify elements common to \( A \), \( B \), and \( C \)**:
- \( A = \{1, 2, 3, 4\} \)
- \( B = \{2, 3, 4\} \)
- \( C = \{1, 3\} \)

2. **Find the common elements**:
- The only element that is present in all three sets is \( 3 \).

3. **Include the empty set**:
- The empty set \( \emptyset \) is also a valid subset.

4. **Conclusion for (iii)**:
- Thus, \( X = \{3\} \) and \( X = \emptyset \).

### Final Results:
- (i) \( X = \{3\}, \emptyset \)
- (ii) \( X = \emptyset, \{2\}, \{4\}, \{2, 4\} \)
- (iii) \( X = \{3\}, \emptyset \)