If p and q are two statements , then which of the following statements is not equivalent to `phArr(prArrq)` ?

Options

A. `p^^q`
B. `(phArrq)^^(pvvq)`
C. `(prArrq)hArrq`
D. `(-prArrq)^^(pvv~q)`

Answer

To determine which statement is not equivalent to \( p \iff (p \implies q) \), we will analyze the logical expressions step by step.

### Step 1: Understand the expressions
The expression \( p \iff (p \implies q) \) can be broken down into its components:
- \( p \implies q \) means "if \( p \) is true, then \( q \) is true". This is only false when \( p \) is true and \( q \) is false.
- \( p \iff (p \implies q) \) means "both \( p \) and \( p \implies q \) have the same truth value".

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

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

From the truth table:
- \( p \iff (p \implies q) \) is true when \( p \) is false (regardless of \( q \)) or when both \( p \) and \( q \) are true.

### Step 3: Analyze the options
We need to check the equivalence of \( p \iff (p \implies q) \) with other logical statements.

1. **Option 1**: \( p \land q \) (Conjunction)
- Truth table:
| \( p \) | \( q \) | \( p \land q \) |
|---------|---------|------------------|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
- This is not equivalent.

2. **Option 2**: \( p \lor q \) (Disjunction)
- Truth table:
| \( p \) | \( q \) | \( p \lor q \) |
|---------|---------|-----------------|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
- This is not equivalent.

3. **Option 3**: \( p \implies q \)
- Truth table:
| \( p \) | \( q \) | \( p \implies q \) |
|---------|---------|---------------------|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
- This is not equivalent.

4. **Option 4**: \( \neg p \lor q \) (Negation and Disjunction)
- Truth table:
| \( p \) | \( q \) | \( \neg p \) | \( \neg p \lor q \) |
|---------|---------|---------------|----------------------|
| T | T | F | T |
| T | F | F | F |
| F | T | T | T |
| F | F | T | T |
- This is equivalent.

### Conclusion
The statement that is **not equivalent** to \( p \iff (p \implies q) \) is **Option 1: \( p \land q \)**.