Which of the following pairs are logically equivalent ?

Options

A. Conditional, Contrapositive
B. Conditional , Inverse
C. Contrapositive, converse
D. Inverse, contrapositive

Answer

To determine which pairs of statements are logically equivalent, we will analyze each option step by step.

### Step 1: Understand the Definitions
1. **Conditional Statement**: A conditional statement is of the form \( p \rightarrow q \) (if \( p \) then \( q \)).
2. **Contrapositive**: The contrapositive of \( p \rightarrow q \) is \( \neg q \rightarrow \neg p \) (if not \( q \) then not \( p \)).
3. **Inverse**: The inverse of \( p \rightarrow q \) is \( \neg p \rightarrow \neg q \) (if not \( p \) then not \( q \)).
4. **Converse**: The converse of \( p \rightarrow q \) is \( q \rightarrow p \) (if \( q \) then \( p \)).

### Step 2: Analyze Each Pair
1. **Conditional and Contrapositive**:
- Conditional: \( p \rightarrow q \)
- Contrapositive: \( \neg q \rightarrow \neg p \)
- These two statements are logically equivalent. (True)

2. **Conditional and Inverse**:
- Conditional: \( p \rightarrow q \)
- Inverse: \( \neg p \rightarrow \neg q \)
- These two statements are not logically equivalent. (False)

3. **Contrapositive and Converse**:
- Contrapositive: \( \neg q \rightarrow \neg p \)
- Converse: \( q \rightarrow p \)
- These two statements are not logically equivalent. (False)

4. **Inverse and Contrapositive**:
- Inverse: \( \neg p \rightarrow \neg q \)
- Contrapositive: \( \neg q \rightarrow \neg p \)
- These two statements are not logically equivalent. (False)

### Conclusion
From the analysis, we find that the only pair that is logically equivalent is the conditional statement and its contrapositive.

### Final Answer
The logically equivalent pair is:
- Conditional Statement and Contrapositive.