https://file.aiquickdraw.com/mathbot/file/1726726763775sqmhkbz8.jpg

The image shows the logical equivalence:

\[
p \rightarrow q \text{ is logically equivalent to } \neg q \rightarrow \neg p
\]

This statement is known as the *contrapositive* in propositional logic. It asserts that a conditional statement \( p \rightarrow q \) (if \( p \), then \( q \)) is logically equivalent to its contrapositive \( \neg q \rightarrow \neg p \) (if not \( q \), then not \( p \)).

### Explanation:

1. **Conditional Statement ( \( p \rightarrow q \) )**: This means "if \( p \) is true, then \( q \) is true."
2. **Contrapositive ( \( \neg q \rightarrow \neg p \) )**: This means "if \( q \) is false, then \( p \) is false."

The contrapositive always holds the same truth value as the original implication, which is why they are logically equivalent.

Would you like further details or examples of how this works?

### Related Questions:
1. What is the converse of \( p \rightarrow q \)?
2. How does the contrapositive differ from the inverse?
3. Can you give a truth table for the equivalence of \( p \rightarrow q \) and \( \neg q \rightarrow \neg p \)?
4. How is the contrapositive used in mathematical proofs?
5. What are some common logical fallacies involving implications?

### Tip:
When proving statements involving implications, considering the contrapositive can often simplify the reasoning.

Is the statement p → q logically equivalent to ¬q → ¬p?

Explore the logical equivalence between the conditional statement p → q and its contrapositive ¬q → ¬p. Learn how contrapositive statements maintain the same truth value as their corresponding conditionals. Suitable for advanced high school and university students.

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