For any two statement p and q `~(pveeq)vee(~pwedgeq)` logically to

Options

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

Answer

To solve the logical expression \( \neg (p \lor q) \lor (\neg p \land q) \), we will use a truth table to evaluate the expression step by step.

### Step-by-Step Solution:

1. **Define the Variables:**
- Let \( p \) and \( q \) be two logical statements. They can either be True (T) or False (F).

2. **Construct the Truth Table:**
- We will create a truth table with all possible combinations of truth values for \( p \) and \( q \).

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

3. **Evaluate Each Column:**
- **Column 3:** \( p \lor q \) is True if at least one of \( p \) or \( q \) is True.
- **Column 4:** \( \neg (p \lor q) \) is the negation of \( p \lor q \).
- **Column 5:** \( \neg p \) is the negation of \( p \).
- **Column 6:** \( \neg p \land q \) is True only if both \( \neg p \) and \( q \) are True.
- **Column 7:** Finally, \( \neg (p \lor q) \lor (\neg p \land q) \) is True if either \( \neg (p \lor q) \) or \( \neg p \land q \) is True.

4. **Final Evaluation:**
- From the last column, we see that the expression \( \neg (p \lor q) \lor (\neg p \land q) \) evaluates to True for the combinations \( (F, T) \) and \( (F, F) \) and False for \( (T, T) \) and \( (T, F) \).
- This corresponds to the truth values of \( \neg p \) when \( p \) is False.

5. **Conclusion:**
- The expression \( \neg (p \lor q) \lor (\neg p \land q) \) is logically equivalent to \( \neg p \).

### Final Answer:
The expression \( \neg (p \lor q) \lor (\neg p \land q) \) is logically equivalent to \( \neg p \).

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