Which of the following is not equivalent from of "if and only if " for the given statements p and q

Options

A. If p then q
B. q if and only if p
C. p if and only if q
D. p is necesssary and suffient condition for q and vice-versa

Answer

To determine which of the following statements is not an equivalent form of "if and only if" (denoted as \( p \iff q \)) for the given statements \( p \) and \( q \), we need to analyze the definitions and implications of the logical statements.

### Step-by-Step Solution:

1. **Understanding "If and Only If":**
The statement \( p \iff q \) means that both \( p \) implies \( q \) and \( q \) implies \( p \). This can be expressed as:
\[ (p \implies q) \land (q \implies p) \]
This means that \( p \) is true if and only if \( q \) is true.

2. **Analyzing the Options:**
We need to evaluate the given options to see which one does not represent the same logical relationship as \( p \iff q \).

- **Option 1:** \( p \implies q \)
- This is a one-way implication, meaning if \( p \) is true, then \( q \) must also be true. However, it does not account for the case where \( q \) could be true without \( p \) being true. Therefore, this is not equivalent to \( p \iff q \).

- **Option 2:** \( q \implies p \)
- Similar to Option 1, this is also a one-way implication. It states that if \( q \) is true, then \( p \) must also be true. This does not capture the bidirectional nature of \( p \iff q \).

- **Option 3:** \( p \text{ is necessary and sufficient for } q \)
- This statement directly reflects the definition of \( p \iff q \) since it indicates that \( p \) being true guarantees \( q \) is true and vice versa. Thus, this is equivalent to \( p \iff q \).

- **Option 4:** \( p = q \) or \( q = p \)
- This statement also captures the essence of \( p \iff q \), as it indicates that \( p \) and \( q \) are equal in truth value.

3. **Conclusion:**
From the analysis, we find that Options 3 and 4 are equivalent to \( p \iff q \), while Options 1 and 2 are not. However, since we are looking for the one that is not equivalent, we can conclude that:

**The answer is Option 1: \( p \implies q \)**.