The Boolean expression `~(p vee q) vee (~p wedge q)` is equivalent to :

Options

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

Answer

To solve the Boolean expression `~(p ∨ q) ∨ (~p ∧ q)`, we will use a truth table to evaluate its equivalence.

### Step 1: Create a Truth Table
We will create a truth table with columns for `p`, `q`, `p ∨ q`, `~(p ∨ q)`, `~p`, `~p ∧ q`, and finally `~(p ∨ q) ∨ (~p ∧ q)`.

| p | q | p ∨ q | ~(p ∨ q) | ~p | ~p ∧ q | ~(p ∨ q) ∨ (~p ∧ q) |
|-------|-------|-------|----------|-------|--------|----------------------|
| True | True | True | False | False | False | False |
| True | False | True | False | False | False | False |
| False | True | True | False | True | True | True |
| False | False | False | True | True | False | True |

### Step 2: Evaluate Each Column
1. **p ∨ q**: This column is true if either `p` or `q` is true.
2. **~(p ∨ q)**: This column is the negation of the previous column.
3. **~p**: This column is the negation of `p`.
4. **~p ∧ q**: This column is true if both `~p` and `q` are true.
5. **~(p ∨ q) ∨ (~p ∧ q)**: This column combines the results of the previous two columns using disjunction (OR).

### Step 3: Analyze the Final Column
From the final column of the truth table, we see the results:
- For (p, q) = (True, True) → False
- For (p, q) = (True, False) → False
- For (p, q) = (False, True) → True
- For (p, q) = (False, False) → True

### Conclusion
The expression `~(p ∨ q) ∨ (~p ∧ q)` evaluates to True when `p` is False and `q` is True or when both `p` and `q` are False. Thus, the equivalent expression is:

**Answer: `~p ∧ q ∨ ~(p ∨ q)` is equivalent to `q ∧ ~p`.**