Which of the following is a tautology

Options

A. `~p^^(pvvq) to q`
B. `~pvv(pvvq) to q`
C. `~pvv(p^^q) to q`
D. None of these

Answer

To determine which of the given expressions is a tautology, we need to analyze each option and check if it evaluates to true for all possible truth values of its variables. A tautology is a logical statement that is always true, regardless of the truth values of its components.

Let's denote the expressions as follows:
1. Option A: \( \neg P \land (P \lor Q) \implies Q \)
2. Option B: \( \neg P \lor (P \lor Q) \implies Q \)
3. Option C: \( \neg P \lor (P \land Q) \implies Q \)

### Step 1: Create a truth table
We will create a truth table for each expression to evaluate their truth values based on all combinations of truth values for \( P \) and \( Q \).

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

### Step 2: Evaluate each expression
Now, we will evaluate the implication for each expression based on the truth table we created.

1. **For Option A: \( \neg P \land (P \lor Q) \implies Q \)**:
- The implication is false only when \( \neg P \land (P \lor Q) \) is true and \( Q \) is false.
- From the truth table, this is false in the last row (F, F).
- Therefore, Option A is **not a tautology**.

2. **For Option B: \( \neg P \lor (P \lor Q) \implies Q \)**:
- The implication is false only when \( \neg P \lor (P \lor Q) \) is true and \( Q \) is false.
- From the truth table, this is false in the last row (F, F).
- Therefore, Option B is **not a tautology**.

3. **For Option C: \( \neg P \lor (P \land Q) \implies Q \)**:
- The implication is false only when \( \neg P \lor (P \land Q) \) is true and \( Q \) is false.
- From the truth table, this is false in the last row (F, F).
- Therefore, Option C is **not a tautology**.

### Conclusion
After evaluating all options, we find that none of the expressions provided is a tautology. However, if we were to find a tautology among other expressions, we would look for an expression that evaluates to true in all cases.