If p and q are two statements, then ` p vv ~ ( p Rightarrow ~ q)` is equivalent to

Options

A. `p ^^q`
B. P
C. q
D. `~ p ^^ q`

Answer

To solve the problem, we need to determine the equivalence of the expression \( p \lor \neg (p \Rightarrow \neg q) \). We'll do this by constructing a truth table.

### Step 1: Understand the components of the expression
The expression consists of:
- \( p \): a statement
- \( q \): another statement
- \( \neg q \): the negation of \( q \)
- \( p \Rightarrow \neg q \): the implication from \( p \) to \( \neg q \)
- \( \neg (p \Rightarrow \neg q) \): the negation of the implication
- \( p \lor \neg (p \Rightarrow \neg q) \): the final expression we want to analyze

### Step 2: Construct the truth table
We will create a truth table to evaluate the expression for all possible truth values of \( p \) and \( q \).

| \( p \) | \( q \) | \( \neg q \) | \( p \Rightarrow \neg q \) | \( \neg (p \Rightarrow \neg q) \) | \( p \lor \neg (p \Rightarrow \neg q) \) |
|---------|---------|---------------|------------------------------|------------------------------------|------------------------------------------|
| T | T | F | F | T | T |
| T | F | T | T | F | T |
| F | T | F | T | F | F |
| F | F | T | T | F | F |

### Step 3: Evaluate each column
1. **Negation of \( q \)**:
- If \( q \) is true (T), then \( \neg q \) is false (F).
- If \( q \) is false (F), then \( \neg q \) is true (T).

2. **Implication \( p \Rightarrow \neg q \)**:
- This is false only when \( p \) is true and \( \neg q \) is false (i.e., \( p \) is T and \( q \) is T).
- In all other cases, it is true.

3. **Negation of the implication**:
- This column is simply the negation of the previous column.

4. **Final expression \( p \lor \neg (p \Rightarrow \neg q) \)**:
- This is true if either \( p \) is true or \( \neg (p \Rightarrow \neg q) \) is true.

### Step 4: Analyze the results
From the truth table, we can see that the final column \( p \lor \neg (p \Rightarrow \neg q) \) has the same truth values as \( p \):
- When \( p \) is true, the expression is true.
- When \( p \) is false, the expression is false.

### Conclusion
Thus, we conclude that:
\[ p \lor \neg (p \Rightarrow \neg q) \text{ is equivalent to } p. \]

### Final Answer
The expression \( p \lor \neg (p \Rightarrow \neg q) \) is equivalent to \( p \).