The proposition `~(p vv ~q) vv ~(p vv q)` is logically equivalent to

Options

A. p
B. q
C. `~p`
D. `~q`

Answer

To determine the logical equivalence of the proposition `~(p ∨ ~q) ∨ ~(p ∨ q)`, we will use a truth table. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Variables
We have two variables, \( p \) and \( q \).

### Step 2: Create a Truth Table
We will create a truth table that includes all possible combinations of truth values for \( p \) and \( q \).

| \( p \) | \( q \) | \( \sim q \) | \( p \lor \sim q \) | \( p \lor q \) | \( \sim (p \lor \sim q) \) | \( \sim (p \lor q) \) | \( \sim (p \lor \sim q) \lor \sim (p \lor q) \) |
|---------|---------|---------------|----------------------|----------------|-----------------------------|-----------------------|--------------------------------------------------|
| T | T | F | T | T | F | F | F |
| T | F | T | T | T | F | F | F |
| F | T | F | F | T | T | F | T |
| F | F | T | T | F | F | T | T |

### Step 3: Fill in the Truth Table
1. **Negation of \( q \)**: \( \sim q \) is true when \( q \) is false.
2. **Disjunction \( p \lor \sim q \)**: This is true if either \( p \) is true or \( \sim q \) is true.
3. **Disjunction \( p \lor q \)**: This is true if either \( p \) is true or \( q \) is true.
4. **Negation of \( (p \lor \sim q) \)**: This is true if \( p \lor \sim q \) is false.
5. **Negation of \( (p \lor q) \)**: This is true if \( p \lor q \) is false.
6. **Final Expression**: We evaluate \( \sim (p \lor \sim q) \lor \sim (p \lor q) \).

### Step 4: Analyze the Results
From the truth table, we see the final column for \( \sim (p \lor \sim q) \lor \sim (p \lor q) \) yields:
- F, F, T, T

### Step 5: Determine Logical Equivalence
The final results indicate that the expression is logically equivalent to \( \sim p \) because:
- The expression is true when \( p \) is false (which is the definition of \( \sim p \)).

### Conclusion
Thus, the proposition `~(p ∨ ~q) ∨ ~(p ∨ q)` is logically equivalent to \( \sim p \).