Negation of `p to (p vee ~q)` is

Options

A. `~p to (~p vee q)`
B. `p wedge (~p wedge q)`
C. `~p vee (~p vee ~q)`
D. `~p to (~p to q)`

Answer

To find the negation of the statement \( p \to (p \vee \neg q) \), we can follow these steps:

### Step 1: Understand the Implication
The statement \( p \to (p \vee \neg q) \) can be rewritten using the equivalence \( A \to B \equiv \neg A \vee B \). Here, \( A \) is \( p \) and \( B \) is \( (p \vee \neg q) \).

### Step 2: Rewrite the Implication
Using the equivalence, we rewrite the statement:
\[ p \to (p \vee \neg q) \equiv \neg p \vee (p \vee \neg q) \]

### Step 3: Simplify the Expression
Now, we can simplify \( \neg p \vee (p \vee \neg q) \):
\[ \neg p \vee p \vee \neg q \]
By the law of excluded middle, \( \neg p \vee p \) is always true (T), so we can simplify it to:
\[ T \vee \neg q \equiv T \]
Thus, the statement \( p \to (p \vee \neg q) \) simplifies to \( T \).

### Step 4: Find the Negation
Now we need to find the negation of the entire statement:
\[ \neg (p \to (p \vee \neg q)) \equiv \neg T \]
Since the negation of true is false, we have:
\[ \neg T \equiv F \]

### Conclusion
Therefore, the negation of \( p \to (p \vee \neg q) \) is:
\[ F \]