If A. B. C are any three arbitrary events, then which one of the following expressions shows that both A and B occur but not C?

Options

A. `A nn barB nn barC`
B. `A nn B nn barC`
C. `A nn B nn C`
D. `A nn barB nn C`

Answer

To solve the problem, we need to express the scenario where both events A and B occur, but event C does not occur.

### Step-by-Step Solution:

1. **Understanding the Events**:
- We have three events: A, B, and C.
- We want to find an expression that indicates A and B occur together, while C does not occur.

2. **Using Intersection for Occurrence**:
- The occurrence of both A and B can be represented using the intersection symbol (∩).
- Therefore, the expression for A and B occurring together is:
\[ A \cap B \]

3. **Using Complement for Non-occurrence**:
- To indicate that event C does not occur, we use the complement of C, denoted as \( C' \) or \( C^c \).
- Thus, the expression for C not occurring is:
\[ C' \]

4. **Combining the Expressions**:
- To express that both A and B occur while C does not occur, we combine the two expressions using the intersection:
\[ A \cap B \cap C' \]

5. **Identifying the Correct Option**:
- Now, we need to check the given options to find the one that matches our derived expression \( A \cap B \cap C' \).
- The correct option from the provided choices is:
\[ \text{Option B: } A \cap B \cap C' \]

### Final Answer:
The expression that shows that both A and B occur but not C is:
\[ A \cap B \cap C' \]