Which of the following is always false ?

Options

A. `(p implies q) -= ~ p implies ~p`
B. `~ (p vv q) -= ~ p vv ~q`
C. `~ (p implies q )-= p ^^ ~q`
D. `~ (p vv q) -= ~ p ^^ ~q`

Answer

To determine which of the given statements is always false, we will analyze each option step by step.

### Step 1: Analyze Option 1
**Statement:** \( p \Rightarrow q \) is equivalent to \( \neg p \Rightarrow \neg p \).

**Truth Table:**
1. Create a truth table for \( p \) and \( q \):
- \( p: T, T, F, F \)
- \( q: T, F, T, F \)

2. Calculate \( p \Rightarrow q \):
- \( T \Rightarrow T = T \)
- \( T \Rightarrow F = F \)
- \( F \Rightarrow T = T \)
- \( F \Rightarrow F = T \)

So, \( p \Rightarrow q \) gives us: \( T, F, T, T \).

3. Calculate \( \neg p \):
- \( \neg p: F, F, T, T \)

4. Now calculate \( \neg p \Rightarrow \neg p \):
- \( F \Rightarrow F = T \)
- \( F \Rightarrow F = T \)
- \( T \Rightarrow T = T \)
- \( T \Rightarrow T = T \)

So, \( \neg p \Rightarrow \neg p \) gives us: \( T, T, T, T \).

5. Compare the results:
- \( p \Rightarrow q: T, F, T, T \)
- \( \neg p \Rightarrow \neg p: T, T, T, T \)

Since the two results are not equivalent, **Option 1 is false**.

### Step 2: Analyze Option 2
**Statement:** \( \neg (p \lor q) \) is equivalent to \( \neg p \lor \neg q \).

This statement is a direct application of De Morgan's Law, which states that:
\[ \neg (p \lor q) \equiv \neg p \land \neg q \]
Thus, **Option 2 is false**.

### Step 3: Analyze Option 3
**Statement:** \( \neg (p \lor q) \) is equivalent to \( \neg p \land \neg q \).

This is again a direct application of De Morgan's Law, which is true. Therefore, **Option 3 is true**.

### Step 4: Analyze Option 4
**Statement:** \( \neg (p \lor q) \) is equivalent to \( \neg p \land \neg q \).

This is also a direct application of De Morgan's Law, which is true. Therefore, **Option 4 is true**.

### Conclusion
Based on our analysis:
- **Option 1 is false.**
- **Option 2 is false.**
- **Option 3 is true.**
- **Option 4 is true.**

Thus, the statements that are always false are **Option 1 and Option 2**.