` ~ ( p hArr q)` is-

Options

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

Answer

To determine the expression \( \sim (p \leftrightarrow q) \), we will construct a truth table and analyze the logical equivalences step by step.

### Step 1: Define the Variables
We have two variables, \( p \) and \( q \). They can take values of either True (T) or False (F).

### Step 2: Create the Truth Table
We will create a truth table for \( p \) and \( q \):

| \( p \) | \( q \) |
|---------|---------|
| T | T |
| T | F |
| F | T |
| F | F |

### Step 3: Calculate \( p \leftrightarrow q \)
The biconditional \( p \leftrightarrow q \) is true if both \( p \) and \( q \) have the same truth value. We can fill this in the truth table:

| \( p \) | \( q \) | \( p \leftrightarrow q \) |
|---------|---------|----------------------------|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |

### Step 4: Negate \( p \leftrightarrow q \)
Now we need to find \( \sim (p \leftrightarrow q) \). This means we will negate the values in the column for \( p \leftrightarrow q \):

| \( p \) | \( q \) | \( p \leftrightarrow q \) | \( \sim (p \leftrightarrow q) \) |
|---------|---------|----------------------------|-----------------------------------|
| T | T | T | F |
| T | F | F | T |
| F | T | F | T |
| F | F | T | F |

### Step 5: Final Result
The final column \( \sim (p \leftrightarrow q) \) gives us the truth values for the negation of the biconditional. The results are:

- For \( (T, T) \): F
- For \( (T, F) \): T
- For \( (F, T) \): T
- For \( (F, F) \): F

Thus, \( \sim (p \leftrightarrow q) \) is equivalent to the expression that is true when \( p \) and \( q \) have different truth values (i.e., it is equivalent to \( p \oplus q \), the exclusive OR).