The statement `~(phArr q)` is not equivalent to

Options

A. `~phArr q`
B. `(p^^ ~q) vv (q ^^ ~q)`
C. `(p vv q) ^^ (~p vv ~q)`
D. `(p vv q) rArr (p ^^ q)`

Answer

To solve the problem of determining which statement is not equivalent to the negation of \( p \) double implication \( q \) (denoted as \( \sim (p \iff q) \)), we will construct truth tables for the expressions involved and compare them.

### Step 1: Construct the truth table for \( p \) and \( q \)

We will consider all possible truth values for \( p \) and \( q \). Since both \( p \) and \( q \) can be either true (T) or false (F), we will have the following combinations:

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

### Step 2: Calculate \( p \iff q \)

The double implication \( p \iff q \) is true if both \( p \) and \( q \) are either true or both are false. We can fill this in the truth table:

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

### Step 3: Calculate \( \sim (p \iff q) \)

Now we will negate the results of \( p \iff q \):

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

### Step 4: Analyze the options

We need to compare \( \sim (p \iff q) \) with the options provided. Let's denote the options as follows:

1. \( p \land \sim q \)
2. \( \sim p \lor \sim q \)
3. \( p \lor \sim q \)
4. \( p \land q \)

We will construct truth tables for each of these options.

#### Option 1: \( p \land \sim q \)

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

#### Option 2: \( \sim p \lor \sim q \)

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

#### Option 3: \( p \lor \sim q \)

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

#### Option 4: \( p \land q \)

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

### Step 5: Compare the results

Now we compare the results of \( \sim (p \iff q) \) with each option:

- \( \sim (p \iff q) \): F, T, T, F
- Option 1: F, T, F, F
- Option 2: F, T, T, T
- Option 3: T, T, F, T
- Option 4: T, F, F, F

### Conclusion

The only option that does not match the truth values of \( \sim (p \iff q) \) is **Option 2**: \( \sim p \lor \sim q \).