The negation of ` A to (A vv ~ B) `is

Options

A. a fallacy
B. a tautology
C. equivalent to ` ( A vv ~ B) to A`
D. equivalent to ` A to ( A ^^ ~ B)`

Explanation

To find the negation of the expression \( A \to (A \lor \neg B) \), we will follow a systematic approach using logical equivalences and truth tables.

### Step-by-Step Solution:

1. **Identify the Expression**:
We need to negate the expression \( A \to (A \lor \neg B) \).

2. **Rewrite the Implication**:
Recall that an implication \( P \to Q \) can be rewritten as \( \neg P \lor Q \). Therefore, we can rewrite our expression as:
\[ A \to (A \lor \neg B) \equiv \neg A \lor (A \lor \neg B) \]

3. **Simplify the Expression**:
We can simplify \( \neg A \lor (A \lor \neg B) \) using the associative and commutative properties of disjunction:
\[ \neg A \lor A \lor \neg B \]
According to the law of excluded middle, \( \neg A \lor A \) is always true (T). Thus, we have:
\[ T \lor \neg B \equiv T \]
This means that the expression \( A \to (A \lor \neg B) \) is always true regardless of the values of \( A \) and \( B \).

4. **Negate the Expression**:
Now, we need to negate the entire expression:
\[ \neg (A \to (A \lor \neg B)) \equiv \neg T \]
Since the negation of true (T) is false (F), we have:
\[ \neg T \equiv F \]

5. **Conclusion**:
Therefore, the negation of \( A \to (A \lor \neg B) \) is:
\[ \text{False (F)} \]

### Final Answer:
The negation of \( A \to (A \lor \neg B) \) is **False (F)**.