Which of the following statement is a tautology?

Options

A. `(p vv r) vv ~ (p ^^ q)`
B. `(~ p ^^ q) vv (~ p vv q)`
C. `~ [(p vv q) ^^ ~ q]`
D. `(p to q) ^^ (q to r)`

Explanation

To determine which of the given statements is a tautology, we need to analyze each option using truth tables. A tautology is a statement that is always true, regardless of the truth values of its components.

Let's denote the statements as follows:
1. \( S_1: P \lor R \)
2. \( S_2: \neg P \land Q \lor \neg P \)
3. \( S_3: (P \lor Q) \land \neg Q \)
4. \( S_4: P \implies Q \land Q \implies R \)

### Step 1: Create a Truth Table
We will create a truth table for the variables \( P \), \( Q \), and \( R \). The possible truth values are True (T) and False (F).

| P | Q | R | \( P \lor R \) | \( \neg P \) | \( \neg P \land Q \) | \( S_2 \) | \( P \lor Q \) | \( \neg Q \) | \( S_3 \) | \( P \implies Q \) | \( Q \implies R \) | \( S_4 \) |
|-------|-------|-------|-----------------|---------------|-----------------------|-----------|-----------------|---------------|-----------|---------------------|---------------------|-----------|
| T | T | T | T | F | F | F | T | F | F | T | T | T |
| T | T | F | T | F | F | F | T | F | F | T | F | F |
| T | F | T | T | F | F | F | T | T | F | T | T | T |
| T | F | F | T | F | F | F | T | T | F | T | F | F |
| F | T | T | T | T | T | T | T | F | F | F | T | F |
| F | T | F | F | T | T | T | T | F | F | F | F | F |
| F | F | T | T | T | F | F | F | T | F | F | T | F |
| F | F | F | F | T | F | F | F | T | F | F | F | F |

### Step 2: Evaluate Each Statement
Now we will evaluate each statement based on the truth table:

1. **For \( S_1: P \lor R \)**:
- The truth values are T, T, T, T, T, F, T, F.
- Not a tautology (has F).

2. **For \( S_2: \neg P \land Q \lor \neg P \)**:
- The truth values are F, F, F, F, T, T, F, F.
- Not a tautology (has F).

3. **For \( S_3: (P \lor Q) \land \neg Q \)**:
- The truth values are F, F, F, F, F, F, F, F.
- Not a tautology (all F).

4. **For \( S_4: P \implies Q \land Q \implies R \)**:
- The truth values are T, F, T, F, F, F, F, F.
- Not a tautology (has F).

### Conclusion
None of the statements provided are tautologies as they do not yield a truth value of True in all cases.