Which of the following is true for the propositions p and q ?

Options

A. p ^ q is true when at least one of p and q is true
B. ` p to q ` is true when p is true and q is false
C. `p harr q ` is true only when both p and q are ture
D. ` ~ ( p vv q)` is true only when both p and q are false

Explanation

To determine which of the given propositions about p and q is true, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding the Propositions**: We have two propositions, p and q. We need to evaluate four options to see which one is true based on the logical relationships between p and q.

2. **Option 1**: "P and Q is true when at least one of P and Q is true."
- This statement is incorrect. The conjunction (AND operation) P and Q is true only when both p and q are true. If at least one is true, it does not guarantee that P and Q is true. Therefore, this option is **false**.

3. **Option 2**: "If P then Q is true when P is true and Q is false."
- This statement is also incorrect. The implication (P → Q) is false only when P is true and Q is false. Therefore, this option is **false**.

4. **Option 3**: "P if and only if Q is true only when both P and Q are true."
- This statement is misleading. The biconditional (P ↔ Q) is true when both p and q are true or both are false. Thus, this option is **false**.

5. **Option 4**: "The negation of P and Q is true only when both P and Q are false."
- According to De Morgan's Theorem, the negation of (P and Q) is equivalent to (not P) or (not Q). This statement is true when both p and q are false, making the negation true. Therefore, this option is **true**.

### Conclusion:
The correct answer is **Option 4**.