The false statement among the following is

Options

A. `p^^(~p)` is a contradiction
B. `(p to q) harr (~ q to ~p)` is a contradiction
C. `pvv(~p)` is a tautology
D. none of these

Explanation

To determine the false statement among the given options, we will analyze each option using truth tables. Let's denote the statements as follows:

1. **Option 1**: \( P \land \neg P \) (P and not P)
2. **Option 2**: \( P \implies Q \equiv (\neg P \lor Q) \) (P implies Q is equivalent to not P or Q)
3. **Option 3**: \( P \lor \neg P \) (P or not P)

### Step 1: Analyze Option 1
We will create a truth table for \( P \land \neg P \).

| P | ¬P | P ∧ ¬P |
|-------|-------|--------|
| True | False | False |
| False | True | False |

**Conclusion for Option 1**: The statement \( P \land \neg P \) is always false, which means it is a contradiction.

### Step 2: Analyze Option 2
Next, we will analyze \( P \implies Q \equiv (\neg P \lor Q) \) using a truth table.

| P | Q | ¬P | P → Q | ¬P ∨ Q |
|-------|-------|-------|-------|--------|
| True | True | False | True | True |
| True | False | False | False | False |
| False | True | True | True | True |
| False | False | True | True | True |

**Conclusion for Option 2**: The implication \( P \implies Q \) is equivalent to \( \neg P \lor Q \). This statement is not always false; it can be true or false depending on the values of P and Q.

### Step 3: Analyze Option 3
Now, we will analyze \( P \lor \neg P \).

| P | ¬P | P ∨ ¬P |
|-------|-------|--------|
| True | False | True |
| False | True | True |

**Conclusion for Option 3**: The statement \( P \lor \neg P \) is always true. This means it is a tautology.

### Final Conclusion
From our analysis:
- **Option 1**: is a contradiction (always false).
- **Option 2**: is not always false; it can be true or false.
- **Option 3**: is a tautology (always true).

Thus, the false statement among the options is **Option 1**.