` ~ [ ~ p ^^ ( p harr q)] -=` is equivalent to

Options

A. ` p vv q`
B. ` q ^^ q`
C. T
D. F

Answer

To solve the problem, we need to simplify the given expression \( \sim [ \sim p \land (p \leftrightarrow q)] \) and find its equivalent form.

### Step-by-Step Solution:

1. **Identify the Expression**:
The expression we need to simplify is \( \sim [ \sim p \land (p \leftrightarrow q)] \).

2. **Apply De Morgan's Law**:
According to De Morgan's Law, the negation of a conjunction can be expressed as the disjunction of the negations:
\[ \sim (A \land B) = \sim A \lor \sim B \]
Here, let \( A = \sim p \) and \( B = (p \leftrightarrow q) \). Therefore, we can rewrite the expression as:
\[ \sim [\sim p \land (p \leftrightarrow q)] = \sim (\sim p) \lor \sim (p \leftrightarrow q) \]

3. **Simplify \( \sim (\sim p) \)**:
The negation of the negation of \( p \) gives us:
\[ \sim (\sim p) = p \]
So the expression now becomes:
\[ p \lor \sim (p \leftrightarrow q) \]

4. **Simplify \( \sim (p \leftrightarrow q) \)**:
The biconditional \( p \leftrightarrow q \) is true when both \( p \) and \( q \) are either true or false. Thus, its negation can be expressed as:
\[ \sim (p \leftrightarrow q) = p \oplus q \]
where \( \oplus \) denotes the exclusive OR (XOR). Therefore, we can rewrite our expression as:
\[ p \lor (p \oplus q) \]

5. **Final Simplification**:
The expression \( p \lor (p \oplus q) \) can be simplified further. The exclusive OR \( p \oplus q \) is true if either \( p \) is true and \( q \) is false, or \( p \) is false and \( q \) is true. Therefore, the overall expression evaluates to true if \( p \) is true or if \( p \) is false and \( q \) is true.

6. **Truth Table Verification**:
To confirm our result, we can create a truth table for \( p \lor (p \oplus q) \) and compare it with \( p \lor q \):
- If \( p \) is true, the expression is true regardless of \( q \).
- If \( p \) is false, the expression is true if \( q \) is true.

This matches the truth table for \( p \lor q \).

### Conclusion:
Thus, the expression \( \sim [ \sim p \land (p \leftrightarrow q)] \) is equivalent to \( p \lor q \).