The Boollean expression `:`
`((p ^^ q) vv ( p vv ~ q ) ) ^^ ( ~ p ^^ ~ q )` is equivalent to `:`

Options

A. `p ^^ q`
B. `( ~ p ) ^^ ( ~q)`
C. `p ^^ ( ~ q )`
D. `p vv ( ~ q )`

Answer

To solve the Boolean expression \(((p \land q) \lor (p \lor \neg q)) \land (\neg p \land \neg q)\) and determine its equivalence, we can use a truth table approach. Here are the steps:

### Step 1: Define the Variables
We have two variables, \(p\) and \(q\). We will evaluate the expression for all possible combinations of truth values for \(p\) and \(q\).

### Step 2: Create the Truth Table
We will create a truth table with columns for \(p\), \(q\), \(\neg p\), \(\neg q\), \(p \land q\), \(p \lor \neg q\), and the final expression.

| \(p\) | \(q\) | \(\neg p\) | \(\neg q\) | \(p \land q\) | \(p \lor \neg q\) | \((p \land q) \lor (p \lor \neg q)\) | \(\neg p \land \neg q\) | Final Expression |
|-------|-------|------------|------------|----------------|--------------------|---------------------------------------|-------------------------|------------------|
| T | T | F | F | T | T | T | F | F |
| T | F | F | T | F | T | T | F | F |
| F | T | T | F | F | T | T | F | F |
| F | F | T | T | F | T | T | T | T |

### Step 3: Evaluate Each Column
1. **Negations**:
- \(\neg p\) is the opposite of \(p\).
- \(\neg q\) is the opposite of \(q\).

2. **AND Operation**:
- \(p \land q\) is true only when both \(p\) and \(q\) are true.

3. **OR Operation**:
- \(p \lor \neg q\) is true if either \(p\) is true or \(\neg q\) is true.

4. **Combining Results**:
- The expression \((p \land q) \lor (p \lor \neg q)\) is true if either \(p \land q\) or \(p \lor \neg q\) is true.

5. **Final Expression**:
- The final expression combines the results of \((p \land q) \lor (p \lor \neg q)\) with \(\neg p \land \neg q\) using AND.

### Step 4: Analyze the Final Column
From the truth table, we see that the final expression is true only when both \(p\) and \(q\) are false. This means the expression is equivalent to \(\neg p \land \neg q\).

### Conclusion
The Boolean expression \(((p \land q) \lor (p \lor \neg q)) \land (\neg p \land \neg q)\) is equivalent to \(\neg p \land \neg q\).

---