The compound statement `(phArr q)vv(p hArr ~q)` is logically equivalent to

Options

A. `phArrq`
B. `pvvq`
C. tautology
D. contradiction

Answer

To determine the logical equivalence of the compound statement \( (P \implies Q) \lor (P \iff \neg Q) \), we will use a truth table. Here’s a step-by-step solution:

### Step 1: Set up the truth table
We need to create a truth table that includes columns for \( P \), \( Q \), \( \neg Q \), \( P \implies Q \), \( P \iff \neg Q \), and finally \( (P \implies Q) \lor (P \iff \neg Q) \).

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

### Step 2: Fill in the truth values
- For \( P \) and \( Q \), we have the combinations: (T, T), (T, F), (F, T), (F, F).
- \( \neg Q \) is simply the negation of \( Q \).
- \( P \implies Q \) is true unless \( P \) is true and \( Q \) is false.
- \( P \iff \neg Q \) is true if both \( P \) and \( \neg Q \) are the same (both true or both false).

### Step 3: Evaluate \( (P \implies Q) \lor (P \iff \neg Q) \)
- For each row, we take the logical OR of the values in the columns \( P \implies Q \) and \( P \iff \neg Q \).

### Step 4: Analyze the results
From the truth table, we see that the last column \( (P \implies Q) \lor (P \iff \neg Q) \) is true for all combinations of \( P \) and \( Q \). This means that the compound statement is a tautology.

### Conclusion
The compound statement \( (P \implies Q) \lor (P \iff \neg Q) \) is logically equivalent to a tautology.

### Final Answer
The correct option is **C. Tautology**.