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 \( p \iff q \)), we can follow these steps:

### Step 1: Understand the Biconditional
The biconditional \( p \iff q \) means that \( p \) is true if and only if \( q \) is true. This can be expressed in terms of implications:
\[ p \iff q \text{ is equivalent to } (p \implies q) \land (q \implies p) \]

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

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

### Step 3: Analyze the Truth Table
From the truth table, we can see:
- \( p \iff q \) is true when both \( p \) and \( q \) are true or when both are false.
- \( p \implies q \) is false only when \( p \) is true and \( q \) is false.
- \( q \implies p \) is false only when \( q \) is true and \( p \) is false.

### Step 4: Combine the Implications
The biconditional \( p \iff q \) can be expressed as:
\[ p \iff q \equiv (p \implies q) \land (q \implies p) \]
This means that \( p \iff q \) is logically equivalent to the conjunction of the two implications.

### Conclusion
Thus, the logically equivalent proposition of \( p \iff q \) is:
\[ p \iff q \equiv (p \implies q) \land (q \implies p) \]