Which of the following statements is false when p is true and q is false?

Options

A. `(prArrq)hArrr`
B. `(hArrq)rArr r`
C. `(qrArrr)rArrp`
D. `(r rArr p)rArrq`

Answer

To determine which statement is false when \( p \) is true and \( q \) is false, we will evaluate each of the given logical statements step by step.

### Given:
- \( p = \text{True} \)
- \( q = \text{False} \)

### Steps to Evaluate Each Statement:

1. **Identify the Statements**:
Let's assume the statements are as follows (based on the context):
- A: \( p \implies q \) and \( r \)
- B: \( p \implies q \) biconditional \( q \) and \( r \)
- C: \( q \implies r \implies p \)
- D: \( r \implies p \implies q \)

2. **Evaluate Each Statement**:
We will evaluate each statement based on the truth values of \( p \) and \( q \).

**Statement A: \( p \implies q \) and \( r \)**
- \( p \implies q \): True implies False = False
- Therefore, \( A = \text{False} \) (since False and anything is False).

**Statement B: \( p \implies q \) biconditional \( q \) and \( r \)**
- \( p \implies q \): True implies False = False
- \( q \) is False.
- The biconditional (False biconditional False) = True.
- Therefore, \( B = \text{False} \) (since False and True = False).

**Statement C: \( q \implies r \implies p \)**
- We need to evaluate \( q \implies r \) first.
- Since \( q \) is False, \( q \implies r \) is True (False implies anything is True).
- Now, \( r \implies p \): True implies True = True.
- Therefore, \( C = \text{True} \).

**Statement D: \( r \implies p \implies q \)**
- We need to evaluate \( r \implies p \) first.
- Assuming \( r \) is True, \( r \implies p \) is True (True implies True = True).
- Now, \( p \implies q \): True implies False = False.
- Therefore, \( D = \text{False} \).

3. **Conclusion**:
After evaluating all statements, we find that:
- Statement A is False.
- Statement B is False.
- Statement C is True.
- Statement D is False.

Thus, the statements that are false when \( p \) is true and \( q \) is false are A, B, and D.

### Final Answer:
The false statements are A, B, and D.