To prove that a conditional statement is logically equivalent to its contrapositive, let's break down both statements clearly and show how they are related.

1. Conditional Statement:

A conditional statement is of the form:

$P \implies Q$

This means "if P, then Q." In other words, if the proposition $P$ is true, then $Q$ must also be true.

2. Contrapositive Statement:

The contrapositive of a conditional statement $P \implies Q$ is the statement:

$\neg Q \implies \neg P$

This means "if not Q, then not P." In other words, if $Q$ is false, then $P$ must also be false.

3. Logical Equivalence:

To show that $P \implies Q$ is logically equivalent to $\neg Q \implies \neg P$, we need to demonstrate that both statements have the same truth value for all possible truth values of $P$ and $Q$.

Truth Tables for $P \implies Q$ and $\neg Q \implies \neg P$:

$P$	$Q$	$P \implies Q$	$\neg Q$	$\neg P$	$\neg Q \implies \neg P$
T	T	T	F	F	T
T	F	F	T	F	F
F	T	T	F	T	T
F	F	T	T	T	T

• $P \implies Q$ is true in all cases except when $P$ is true and $Q$ is false.
• $\neg Q \implies \neg P$ is true in all cases except when $\neg Q$ is true (meaning $Q$ is false) and $\neg P$ is false (meaning $P$ is true).

From the table, we see that both $P \implies Q$ and $\neg Q \implies \neg P$ have the same truth values for all possible combinations of $P$ and $Q$. Therefore, we conclude that the conditional statement is logically equivalent to its contrapositive.

---

Would you like further clarification on any step or examples? Here are some follow-up questions to explore:

1. What is the difference between a converse and a contrapositive?
2. How do logical equivalences apply to proofs and mathematical reasoning?
3. What is the truth table for the converse of a conditional statement?
4. How does the contrapositive play a role in proof by contradiction?
5. Can you explain other common logical equivalences with truth tables?

Tip: A helpful mnemonic for remembering the contrapositive is: "Flip and negate" — flip the positions of the terms in the conditional and negate both of them!