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 determine the equivalence of the expression \( p \lor \neg (p \Rightarrow \neg q) \), we will construct a truth table and analyze the logical relationships step by step.

### Step-by-Step Solution:

1. **Identify the Statements**:
- Let \( p \) and \( q \) be two statements.

2. **Construct the Truth Table**:
- We will create a truth table that includes the columns for \( p \), \( q \), \( \neg q \), \( p \Rightarrow \neg q \), and \( p \lor \neg (p \Rightarrow \neg q) \).

3. **Fill in the Truth Values**:
- The possible truth values for \( p \) and \( q \) are:
- \( T, T \)
- \( T, F \)
- \( F, T \)
- \( F, F \)

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

4. **Evaluate Each Column**:
- **Column for \( \neg q \)**:
- If \( q \) is true (T), then \( \neg q \) is false (F).
- If \( q \) is false (F), then \( \neg q \) is true (T).
- **Column for \( p \Rightarrow \neg q \)**:
- This is true unless \( p \) is true and \( \neg q \) is false.
- **Column for \( \neg (p \Rightarrow \neg q) \)**:
- This is the negation of the previous column.
- **Final Column \( p \lor \neg (p \Rightarrow \neg q) \)**:
- This is true if either \( p \) is true or \( \neg (p \Rightarrow \neg q) \) is true.

5. **Determine the Equivalence**:
- From the final column, we can see that the expression \( p \lor \neg (p \Rightarrow \neg q) \) is equivalent to \( p \) when we compare the truth values.

### Conclusion:
The expression \( p \lor \neg (p \Rightarrow \neg q) \) is equivalent to \( p \).