If ~q `vv` pis F, then which of the following is correct?

Options

A. p`harr` q is T
B. p `rarr` q is T
C. q`rarr` p is T
D. p`rarr`q is F

Answer

To solve the problem, we need to analyze the statement given: "~q ∨ p is F". This means that the negation of q or p is false.

Let's break down the steps to find the correct answer.

### Step 1: Understand the statement
The expression "~q ∨ p" is false. This means that both components of the disjunction must be false for the entire expression to be false.

### Step 2: Set up the conditions
For "~q ∨ p" to be false:
- ~q must be false, which means q must be true.
- p must also be false.

So we have:
- q = True
- p = False

### Step 3: Analyze the options
Now, we need to evaluate the options based on the values we found for p and q.

1. **Option 1: q if and only if p (q ↔ p)**
- This is false because q is True and p is False.

2. **Option 2: p if and only if q (p ↔ q)**
- This is also false because p is False and q is True.

3. **Option 3: p implies q (p → q)**
- This is true because if p is False, then p implies q is always true regardless of the truth value of q.

4. **Option 4: negation of p and q (~(p ∧ q))**
- p ∧ q is False (since p is False), so ~(p ∧ q) is True.

### Step 4: Conclusion
From the analysis, we find that both Option 3 and Option 4 are correct. However, the question asks for which of the following is correct, so we can conclude that at least one of these options is valid.

### Final Answer
The correct options are:
- p implies q (Option 3)
- negation of p and q (Option 4)