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

Options

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

Answer

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

### Step-by-Step Solution:

1. **Evaluate Statement 1: \( p \implies q \) and \( (p \implies q) \iff r \)**
- Given \( p \) is true and \( q \) is false:
- \( p \implies q \) evaluates to \( \text{True} \implies \text{False} \), which is **False**.
- Now, we have \( \text{False} \iff r \). The truth value of this statement depends on \( r \):
- If \( r \) is true, \( \text{False} \iff \text{True} \) is false.
- If \( r \) is false, \( \text{False} \iff \text{False} \) is true.
- Therefore, this statement cannot be definitively classified as false without knowing \( r \).

2. **Evaluate Statement 2: \( p \iff q \implies r \)**
- \( p \iff q \) evaluates to \( \text{True} \iff \text{False} \), which is **False**.
- Now we have \( \text{False} \implies r \):
- A false statement implies anything, so this is true regardless of \( r \).
- Thus, this statement is **not false**.

3. **Evaluate Statement 3: \( q \implies r \implies p \)**
- \( q \implies r \) evaluates to \( \text{False} \implies r \), which is **True** (false implies anything is true).
- Now we have \( \text{True} \implies p \):
- Since \( p \) is true, \( \text{True} \implies \text{True} \) is true.
- Therefore, this statement is also **not false**.

4. **Evaluate Statement 4: \( r \implies p \implies q \)**
- \( r \implies p \) evaluates to \( r \implies \text{True} \), which is **True** (anything implies true is true).
- Now we have \( \text{True} \implies q \):
- Since \( q \) is false, \( \text{True} \implies \text{False} \) is **False**.
- Therefore, this statement is **false**.

### Conclusion:
The only statement that is false when \( p \) is true and \( q \) is false is **Statement 4**.