If p and q are two propositions, then ` ~ ( p harr q)` is

Options

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

Answer

To solve the problem of finding the negation of the double implication \( \sim (p \harr q) \), we will construct a truth table step by step.

### Step 1: Define the Propositions
Let \( p \) and \( q \) be two propositions. The possible truth values for each proposition are True (T) and False (F).

### Step 2: Create the Truth Table
Since there are two propositions, we will have \( 2^2 = 4 \) combinations of truth values for \( p \) and \( q \).

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

### Step 3: Calculate \( p \harr q \)
The double implication \( p \harr q \) is true if both \( p \) and \( q \) are either true or false. We fill in the truth table accordingly.

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

### Step 4: Negate \( p \harr q \)
Next, we need to find \( \sim (p \harr q) \), which is the negation of the values in the \( p \harr q \) column.

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

### Step 5: Final Truth Table
The final truth table shows the negation of \( p \harr q \):

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

### Conclusion
Thus, the negation of the double implication \( \sim (p \harr q) \) evaluates to:
- False when both \( p \) and \( q \) are true or both are false.
- True when one is true and the other is false.