The simplified form of `(p^^q)vv(p^^~q)` is

Options

A. p
B. q
C. `p^^q`
D. `pvvq`

Answer

To simplify the expression \((p \land q) \lor (p \land \neg q)\), we can follow these steps:

### Step 1: Identify the Expression
We start with the expression:
\[ (p \land q) \lor (p \land \neg q) \]

### Step 2: Apply the Distributive Law
According to the distributive law in logic, we can factor out the common term \(p\):
\[ p \land (q \lor \neg q) \]

### Step 3: Simplify the Inner Expression
Next, we simplify the expression \(q \lor \neg q\). This is a tautology, meaning it is always true (1):
\[ q \lor \neg q = 1 \]

### Step 4: Substitute Back
Now, we substitute back into our expression:
\[ p \land 1 \]

### Step 5: Simplify Further
The expression \(p \land 1\) simplifies to just \(p\):
\[ p \]

### Final Answer
Thus, the simplified form of \((p \land q) \lor (p \land \neg q)\) is:
\[ \boxed{p} \]

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