If p is true and q is false, then which of the following statements is NOT true ?

Options

A. `pvvq`
B. `p^^(~q)`
C. `pto p`
D. `ptoq`

Explanation

To solve the problem, we need to analyze the truth values of the given statements based on the conditions that \( P \) is true and \( Q \) is false. Let's go through each option step by step.

### Step-by-Step Solution:

1. **Identify the truth values**:
- Given: \( P \) is true (\( P = T \)) and \( Q \) is false (\( Q = F \)).

2. **Evaluate each statement**:

**Option 1: \( P \neq Q \) (P Distinction Q)**
- The statement \( P \neq Q \) is true if \( P \) and \( Q \) have different truth values.
- Here, \( P = T \) and \( Q = F \), so \( P \neq Q \) is true.
- Conclusion: This statement is true.

**Option 2: \( P \land \neg Q \) (P conjunction negation of Q)**
- First, find \( \neg Q \): Since \( Q \) is false, \( \neg Q \) is true (\( \neg Q = T \)).
- Now evaluate \( P \land \neg Q \): \( P \land \neg Q = T \land T = T \).
- Conclusion: This statement is true.

**Option 3: \( P \implies P \)**
- This is a tautology; it states that if \( P \) is true, then \( P \) is true.
- Since \( P \) is true, \( P \implies P \) is true.
- Conclusion: This statement is true.

**Option 4: \( P \implies Q \)**
- This states that if \( P \) is true, then \( Q \) must also be true.
- Here, \( P = T \) and \( Q = F \), so \( P \implies Q = T \implies F \), which is false.
- Conclusion: This statement is not true.

3. **Final Answer**:
- The statement that is NOT true is **Option 4: \( P \implies Q \)**.