The statement `p to (~q)` is equivalent to

Options

A. `q to p`
B. `~q vv ~ p`
C. `p ^^ ~ q`
D. `~ q to p`

Answer

To determine the equivalent statement for \( p \to \neg q \), we can follow these steps:

### Step 1: Understand the implication
The statement \( p \to \neg q \) can be rewritten using the definition of implication. The implication \( p \to q \) is equivalent to \( \neg p \lor q \). Thus, we can rewrite \( p \to \neg q \) as:
\[ p \to \neg q \equiv \neg p \lor \neg q \]

### Step 2: Apply De Morgan's Law
Next, we can apply De Morgan's Law, which states that \( \neg (A \land B) \equiv \neg A \lor \neg B \) and \( \neg (A \lor B) \equiv \neg A \land \neg B \). In our case, we have:
\[ \neg p \lor \neg q \]
This expression cannot be simplified further using De Morgan's Law, but we can express it in another form.

### Step 3: Rewrite using conjunction
Using the fact that \( \neg p \lor \neg q \) can be expressed as:
\[ \neg (p \land q) \]
This means that \( \neg p \lor \neg q \) is equivalent to saying that it is not the case that both \( p \) and \( q \) are true.

### Step 4: Conclusion
Thus, we conclude that:
\[ p \to \neg q \equiv \neg p \lor \neg q \equiv \neg (p \land q) \]

### Final Answer
The statement \( p \to \neg q \) is equivalent to \( \neg p \lor \neg q \) or \( \neg (p \land q) \).

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