The statement `phArr q` is equal to -

Options

A. `(~p vv q) vv (p vvq)`
B. `(p ^^q) vv (~p^^~q)`
C. `(~p vvq ) ^^ (p vv~q)`
D. `(p ^^q) vv (pvvq)`

Answer

To solve the question regarding the statement "P by conditional Q," we will follow these steps:

### Step 1: Understand the biconditional statement
The biconditional statement \( P \text{ by conditional } Q \) is denoted as \( P \iff Q \). This means that \( P \) is true if and only if \( Q \) is true.

### Step 2: Rewrite the biconditional statement
The biconditional statement can be expressed in terms of implications:
\[ P \iff Q \equiv (P \implies Q) \land (Q \implies P) \]
This means that both implications must hold true for the biconditional to be true.

### Step 3: Express implications in terms of logical operations
The implication \( P \implies Q \) can be rewritten using logical operations:
\[ P \implies Q \equiv \neg P \lor Q \]
Similarly, the implication \( Q \implies P \) can be rewritten as:
\[ Q \implies P \equiv \neg Q \lor P \]

### Step 4: Combine the implications
Now, substituting these expressions back into our biconditional statement, we get:
\[ P \iff Q \equiv (\neg P \lor Q) \land (\neg Q \lor P) \]

### Step 5: Identify the equivalent expression
The expression \( (\neg P \lor Q) \land (\neg Q \lor P) \) is the standard logical form for the biconditional.

### Conclusion
Thus, the statement \( P \text{ by conditional } Q \) is equivalent to:
\[ (\neg P \lor Q) \land (\neg Q \lor P) \]