If `AcapBsubeC` and `AcapB≠phi`. Then which of the following is incorrect (1) (A∪B)∩C ≠ϕ (2) B∩C=ϕ (3) A∩C≠ϕ (4) If (A−B)⊆C, then A⊆C

Options

A. `(AcupB)capCcancel=phi`
B. `BcupC=phi`
C. `AcupCcancel=phi`
D. If `(A-B)subeC,` then `AsubeC`

Answer

To solve the problem, we need to analyze the given options based on the condition that \( A \cap B \neq \emptyset \) (meaning \( A \) and \( B \) have some common elements) and \( A \cap B \subseteq C \) (meaning all elements that are common to \( A \) and \( B \) are also in \( C \)).

Let's go through each option step by step:

### Step 1: Analyze Option (1) \( (A \cup B) \cap C \neq \emptyset \)

- **Explanation**: The union of \( A \) and \( B \) includes all elements that are in either \( A \) or \( B \). Since \( A \cap B \subseteq C \) and \( A \cap B \neq \emptyset \), it follows that there are elements in \( C \) that are also in \( A \) or \( B \). Therefore, the intersection \( (A \cup B) \cap C \) must also contain elements and cannot be empty.
- **Conclusion**: This statement is **true**.

### Step 2: Analyze Option (2) \( B \cap C = \emptyset \)

- **Explanation**: This option states that there are no elements in common between \( B \) and \( C \). However, since \( A \cap B \neq \emptyset \) and \( A \cap B \subseteq C \), there must be at least some elements in \( C \) that are also in \( B \) (specifically, those elements that are in \( A \cap B \)). Thus, \( B \cap C \) cannot be empty.
- **Conclusion**: This statement is **false**.

### Step 3: Analyze Option (3) \( A \cap C \neq \emptyset \)

- **Explanation**: Since \( A \cap B \subseteq C \) and \( A \cap B \neq \emptyset \), it follows that there are elements in \( A \) that are also in \( C \) (specifically, those elements that are in \( A \cap B \)). Therefore, the intersection \( A \cap C \) must contain elements and cannot be empty.
- **Conclusion**: This statement is **true**.

### Step 4: Analyze Option (4) If \( (A - B) \subseteq C \), then \( A \subseteq C \)

- **Explanation**: The statement \( (A - B) \subseteq C \) means that all elements of \( A \) that are not in \( B \) are in \( C \). However, this does not necessarily imply that all elements of \( A \) are in \( C \), since there could be elements in \( A \) that are also in \( B \) (and thus not included in \( A - B \)). Therefore, \( A \) could have elements that are not in \( C \).
- **Conclusion**: This statement is **false**.

### Final Conclusion

From the analysis above, the incorrect options are:

- Option (2) \( B \cap C = \emptyset \) is **false**.
- Option (4) If \( (A - B) \subseteq C \), then \( A \subseteq C \) is also **false**.

Since the question asks for which of the following is incorrect, the answer is:

**Option (2)** is incorrect.