The logically equivalent proposition of ` p harr q` is

Options

A. ` (p ^^ q) vv ( p vv q)`
B. ` (p to q) ^^ ( q to p)`
C. ` ( p to q) vv ( q to p)`
D. ` (p ^^ q) to (p vv q)`

Explanation

To find the logically equivalent proposition of \( p \text{ biconditional } q \) (denoted as \( p \leftrightarrow q \)), we can use a truth table. Here’s a step-by-step solution:

### Step 1: Create a Truth Table for \( p \) and \( q \)

We will list all possible truth values for \( p \) and \( q \). There are four combinations:

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

### Step 2: Determine the Truth Values for \( p \leftrightarrow q \)

The biconditional \( p \leftrightarrow q \) is true when both \( p \) and \( q \) have the same truth value (both true or both false). We can fill in the truth table for \( p \leftrightarrow q \):

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

### Step 3: Analyze the Options for Logical Equivalence

Now we need to check the options provided to see which one matches the truth values of \( p \leftrightarrow q \).

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

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

This does not match.

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

First, we calculate \( p \implies q \) and \( q \implies p \):

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

Now, we combine these using \( \land \):

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

This matches the truth values of \( p \leftrightarrow q \).

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

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

This does not match.

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

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

This does not match.

### Conclusion

The logically equivalent proposition of \( p \leftrightarrow q \) is given by option B: \( (p \implies q) \land (q \implies p) \).

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