The statement `phArr q` is not equivalent to

Options

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

Answer

To solve the problem of determining which statement is not equivalent to \( p \equiv q \), we will construct truth tables for the given options and compare them with the truth table for \( p \equiv q \).

### Step 1: Construct the Truth Table for \( p \equiv q \)

The equivalence \( p \equiv q \) is true if both \( p \) and \( q \) have the same truth values (both true or both false).

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

### Step 2: Analyze Each Option

#### Option A: \( p \lor q \implies p \land q \)

1. Construct the truth table for \( p \lor q \) and \( p \land q \).

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

The result for \( p \lor q \implies p \land q \) is: T, F, F, T.

#### Option B: \( p \land q \implies p \lor q \)

1. Construct the truth table for \( p \land q \) and \( p \lor q \).

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

The result for \( p \land q \implies p \lor q \) is: T, T, T, T.

#### Option C: \( p \lor q \equiv p \land q \)

1. Construct the truth table for \( p \lor q \) and \( p \land q \).

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

The result for \( p \lor q \equiv p \land q \) is: T, F, F, T.

#### Option D: \( \neg (p \lor q) \lor (p \land q) \)

1. Construct the truth table for \( \neg (p \lor q) \) and \( p \land q \).

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

The result for \( \neg (p \lor q) \lor (p \land q) \) is: T, F, F, T.

### Step 3: Compare Results

Now we compare the results of each option with the truth table for \( p \equiv q \):

- \( p \equiv q \): T, F, F, T
- Option A: T, F, F, T (Equivalent)
- Option B: T, T, T, T (Not Equivalent)
- Option C: T, F, F, T (Equivalent)
- Option D: T, F, F, T (Equivalent)

### Conclusion

The statement that is **not equivalent** to \( p \equiv q \) is **Option B**: \( p \land q \implies p \lor q \).