`p vv q` is false when `:`

Options

A. p and q are both true
B. p and q are both false
C. p is true and q is false
D. p is false and q is true

Answer

To determine when the expression \( p \lor q \) (read as "p or q") is false, we can analyze the truth values of \( p \) and \( q \). The logical disjunction \( p \lor q \) is defined as follows:

- \( p \lor q \) is true if at least one of \( p \) or \( q \) is true.
- \( p \lor q \) is false only when both \( p \) and \( q \) are false.

### Step-by-Step Solution:

1. **Understanding the Logical OR**:
The logical OR operation \( p \lor q \) will yield true if either \( p \) is true, \( q \) is true, or both are true.

2. **Truth Table Construction**:
We can create a truth table to analyze all possible combinations of truth values for \( p \) and \( q \):

| \( p \) | \( q \) | \( p \lor q \) |
|---------|---------|-----------------|
| True | True | True |
| True | False | True |
| False | True | True |
| False | False | False |

3. **Identifying When \( p \lor q \) is False**:
From the truth table, we see that \( p \lor q \) is false only in the last row, where both \( p \) and \( q \) are false.

4. **Conclusion**:
Therefore, the expression \( p \lor q \) is false when both \( p \) and \( q \) are false.

### Final Answer:
The expression \( p \lor q \) is false when both \( p \) and \( q \) are false.

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