`~(p hArr q)` is

Options

A. `~p^^~q`
B. `~pvv~q`
C. `(p^^~q) vv (~p ^^q)`
D. None of these

Answer

To solve the problem of finding the negation of the biconditional statement \( \sim(p \leftrightarrow q) \), we can follow these steps:

### Step 1: Understand the biconditional statement
The biconditional statement \( p \leftrightarrow q \) means that both \( p \) and \( q \) are either true or false together. It can be expressed in terms of implications:
\[ p \leftrightarrow q \equiv (p \rightarrow q) \land (q \rightarrow p) \]

### Step 2: Rewrite the implications
Using the implication equivalence, we can rewrite \( p \rightarrow q \) and \( q \rightarrow p \):
\[ p \rightarrow q \equiv \sim p \lor q \]
\[ q \rightarrow p \equiv \sim q \lor p \]
Thus, we can rewrite the biconditional as:
\[ p \leftrightarrow q \equiv (\sim p \lor q) \land (\sim q \lor p) \]

### Step 3: Negate the biconditional statement
Now we need to find the negation of the entire biconditional statement:
\[ \sim(p \leftrightarrow q) \equiv \sim((\sim p \lor q) \land (\sim q \lor p)) \]
Using De Morgan's laws, we can distribute the negation:
\[ \sim(A \land B) \equiv \sim A \lor \sim B \]
where \( A = \sim p \lor q \) and \( B = \sim q \lor p \). Therefore:
\[ \sim(p \leftrightarrow q) \equiv \sim(\sim p \lor q) \lor \sim(\sim q \lor p) \]

### Step 4: Apply De Morgan's laws to each part
Now we apply De Morgan's laws to each part:
1. For \( \sim(\sim p \lor q) \):
\[ \sim(\sim p \lor q) \equiv p \land \sim q \]
2. For \( \sim(\sim q \lor p) \):
\[ \sim(\sim q \lor p) \equiv q \land \sim p \]

### Step 5: Combine the results
Now we combine the results from the two parts:
\[ \sim(p \leftrightarrow q) \equiv (p \land \sim q) \lor (q \land \sim p) \]

### Conclusion
Thus, the negation of the biconditional statement \( \sim(p \leftrightarrow q) \) is:
\[ (p \land \sim q) \lor (q \land \sim p) \]

### Final Answer
The correct option is:
**Option 3: \( p \land \sim q \lor \sim p \land q \)**

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