The false stalement in the following is-

Options

A. `p ^^ (~p)` is a contradiction
B. `(p implies q) hArr (~ q implies ~ p)` is a contradiction
C. ` ~ (~ p) hArr p` is a tautology
D. `p vv (~p)` is a tautology

Answer

To determine which statement is false among the given options, we can use a truth table. Let's denote the statements as follows:

1. \( P \)
2. \( Q \)
3. \( P \implies Q \)
4. \( \neg Q \implies \neg P \)

We will analyze the truth values of these statements step by step.

### Step 1: Construct the Truth Table

We will consider all possible truth values for \( P \) and \( Q \).

| \( P \) | \( Q \) | \( \neg P \) | \( \neg Q \) | \( P \implies Q \) | \( \neg Q \implies \neg P \) |
|---------|---------|---------------|---------------|---------------------|-------------------------------|
| T | T | F | F | T | T |
| T | F | F | T | F | F |
| F | T | T | F | T | T |
| F | F | T | T | T | T |

### Step 2: Analyze \( P \implies Q \)

The implication \( P \implies Q \) is false only when \( P \) is true and \( Q \) is false. From the table, we see that this occurs in the second row.

### Step 3: Analyze \( \neg Q \implies \neg P \)

The implication \( \neg Q \implies \neg P \) is false only when \( \neg Q \) is true and \( \neg P \) is false. This occurs in the second row as well.

### Step 4: Check the Bi-conditional

We need to check if \( P \implies Q \) is equivalent to \( \neg Q \implies \neg P \). We will create a new column for the bi-conditional \( (P \implies Q) \leftrightarrow (\neg Q \implies \neg P) \).

| \( P \) | \( Q \) | \( P \implies Q \) | \( \neg Q \implies \neg P \) | \( (P \implies Q) \leftrightarrow (\neg Q \implies \neg P) \) |
|---------|---------|---------------------|-------------------------------|---------------------------------------------------------------|
| T | T | T | T | T |
| T | F | F | F | T |
| F | T | T | T | T |
| F | F | T | T | T |

From the table, we can see that the bi-conditional is false in the second row, which indicates that \( P \implies Q \) is not equivalent to \( \neg Q \implies \neg P \).

### Conclusion

The false statement among the options is the second one: \( P \implies Q \) is not equivalent to \( \neg Q \implies \neg P \).