`(~pvv~q)` is logically equivalent to

Options

A. ` ~ p wedge ~ q`
B. `p wedge q`
C. `~ (p wedge q)`
D. `p wedge ~ q`

Answer

To determine what the expression \( (\neg p \lor \neg q) \) is logically equivalent to, we can use a truth table. Here’s a step-by-step solution:

### Step 1: Define the Variables
Let \( p \) and \( q \) be two boolean variables. We will evaluate the expression \( \neg p \lor \neg q \).

### Step 2: Create the Truth Table
We will create a truth table for the variables \( p \) and \( q \), their negations \( \neg p \) and \( \neg q \), and the expression \( \neg p \lor \neg q \).

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

### Step 3: Analyze the Truth Table
From the truth table, we can see the output of \( \neg p \lor \neg q \):
- When both \( p \) and \( q \) are true, the output is false.
- When either \( p \) is false or \( q \) is false (or both), the output is true.

### Step 4: Find the Equivalent Expression
Now, we can look for an equivalent expression. The expression \( \neg p \lor \neg q \) can be transformed using De Morgan's Laws:
\[ \neg p \lor \neg q \equiv \neg (p \land q) \]
This means that \( \neg p \lor \neg q \) is logically equivalent to the negation of the conjunction of \( p \) and \( q \).

### Conclusion
Thus, the expression \( (\neg p \lor \neg q) \) is logically equivalent to \( \neg (p \land q) \).