Let `A,B` and `C` be sets such that `Ann B sube C` and `Ann B != phi`. Then which of the following statements is not true?

Options

A. `B nn C ne phi`
B. If `(A-B) sube C`, then `A sube C`
C. `C(uu A) nn (C uu B) = C`
D. If `(A -C) sube B`, then `A sube B`

Answer

To solve the problem, we need to analyze the given conditions and statements regarding the sets \( A \), \( B \), and \( C \).

### Step-by-Step Solution:

1. **Understanding the Given Information:**
We are given that \( A \cap B \subseteq C \) and \( A \cap B \neq \emptyset \). This means that the intersection of sets \( A \) and \( B \) is not only a subset of \( C \) but also contains at least one element.

2. **Visualizing the Sets:**
Let's visualize the sets:
- Let \( A \) and \( B \) be two overlapping sets such that their intersection \( A \cap B \) is not empty.
- Since \( A \cap B \subseteq C \), all elements in the intersection are also in set \( C \).

3. **Analyzing the Statements:**
We need to evaluate which of the following statements is not true based on our understanding:
- **Statement 1:** \( A \cup B \cap C \) is non-empty.
- **Statement 2:** \( B \cap C \) is empty.
- **Statement 3:** \( A \cap C \) is non-empty.
- **Statement 4:** \( A \cap B \) is non-empty.

4. **Evaluating Each Statement:**
- **Statement 1:** \( A \cup B \cap C \) is non-empty.
- Since \( A \cap B \) is non-empty and is a subset of \( C \), \( A \cup B \) will also have elements in \( C \). Thus, this statement is true.

- **Statement 2:** \( B \cap C \) is empty.
- Since \( A \cap B \) is non-empty and \( A \cap B \subseteq C \), it implies that \( B \) must have elements in \( C \). Therefore, this statement is false.

- **Statement 3:** \( A \cap C \) is non-empty.
- Since \( A \cap B \) is non-empty and \( A \cap B \subseteq C \), it follows that \( A \) must have elements in \( C \). Thus, this statement is true.

- **Statement 4:** \( A \cap B \) is non-empty.
- This is given in the problem statement, so it is true.

5. **Conclusion:**
The statement that is not true is **Statement 2: \( B \cap C \) is empty**.

### Final Answer:
The statement that is not true is: **\( B \cap C \) is empty.**