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 analyze the logical equivalence of the given options step by step.

### Step 1: Understand the equivalence \( p \equiv q \)
The statement \( p \equiv q \) (p is equivalent to q) is true when both p and q have the same truth values. This can be summarized in a truth table:

| p | q | \( p \equiv q \) |
|-------|-------|------------------|
| True | True | True |
| True | False | False |
| False | True | False |
| False | False | True |

### Step 2: Analyze the options
We need to evaluate each option to see if it is equivalent to \( p \equiv q \).

#### Option A: \( p \lor q \implies p \land q \)
1. Create a truth table for \( p \lor q \) and \( p \land q \):
- \( p \lor q \) (p or q) is true if at least one of p or q is true.
- \( p \land q \) (p and q) is true only if both p and q are true.

| p | q | \( p \lor q \) | \( p \land q \) | \( p \lor q \implies p \land q \) |
|-------|-------|----------------|------------------|------------------------------------|
| True | True | True | True | True |
| True | False | True | False | False |
| False | True | True | False | False |
| False | False | False | False | True |

The result for \( p \lor q \implies p \land q \) is True, False, False, True, which does not match \( p \equiv q \).

#### Option B: \( p \land q \implies p \lor q \)
1. Create a truth table for \( p \land q \) and \( p \lor q \):

| p | q | \( p \land q \) | \( p \lor q \) | \( p \land q \implies p \lor q \) |
|-------|-------|------------------|----------------|------------------------------------|
| True | True | True | True | True |
| True | False | False | True | True |
| False | True | False | True | True |
| False | False | False | False | True |

The result for \( p \land q \implies p \lor q \) is True, True, True, True, which matches \( p \equiv q \).

#### Option C: \( p \lor q \equiv p \land q \)
1. Create a truth table for \( p \lor q \) and \( p \land q \):

| p | q | \( p \lor q \) | \( p \land q \) | \( p \lor q \equiv p \land q \) |
|-------|-------|----------------|------------------|-----------------------------------|
| True | True | True | True | True |
| True | False | True | False | False |
| False | True | True | False | False |
| False | False | False | False | True |

The result for \( p \lor q \equiv p \land q \) is True, False, False, True, which does not match \( p \equiv q \).

#### Option D: \( p \equiv q \)
This is the original statement, so it is equivalent by definition.

### Conclusion
From the analysis, we find that:
- Option A and Option C are not equivalent to \( p \equiv q \).
- However, the question asks for the one that is not equivalent, and since the options are typically singular, we can conclude that **Option A** is the answer.

### Final Answer
The statement \( p \lor q \implies p \land q \) is not equivalent to \( p \equiv q \).