The Boolean expression `~(pvvq)vv(~p^^q)` is equivalent to (1) `~p` (2) `p` (3) `q` (4) `~q`

Options

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

Answer

To solve the Boolean expression `~(p ∨ q) ∨ (~p ∧ q)`, we will simplify it step by step.

### Step 1: Apply De Morgan's Theorem
The first part of the expression is `~(p ∨ q)`. According to De Morgan's theorem, this can be rewritten as:
\[ ~(p ∨ q) = ~p ∧ ~q \]
So, we can rewrite the expression as:
\[ (~p ∧ ~q) ∨ (~p ∧ q) \]

### Step 2: Factor out the common term
In the expression `(~p ∧ ~q) ∨ (~p ∧ q)`, we can factor out `~p`:
\[ ~p ∧ (~q ∨ q) \]

### Step 3: Simplify using the Law of Excluded Middle
The expression `(~q ∨ q)` is always true (T) because either `q` is true or `q` is false. So we can simplify:
\[ ~p ∧ T = ~p \]

### Conclusion
Thus, the original expression simplifies to:
\[ ~p \]

The equivalent expression is `~p`, which corresponds to option (1).