Contrapositive

Definition

Given a conditional statement “If $P$, then $Q$” ($P \to Q$), the contrapositive is the statement formed by negating and interchanging both the hypothesis ($P$) and the conclusion ($Q$): $\sim Q \to \sim P$ How to read: “Not Q implies not P.” Meaning / when to use: The statement that if the conclusion $Q$ is false, then the hypothesis $P$ must also be false. The contrapositive is logically equivalent to the original conditional statement.

Why It Matters

Because a statement and its contrapositive always have the same truth value, proving the contrapositive is a standard method of indirect proof (proof by contraposition) in mathematics and computer science when the direct statement is difficult to prove.

Core Concepts

• Negate and Swap: Both negates the terms and swaps their positions.
• Logical Equivalence: If $P \to Q$ is true, $\sim Q \to \sim P$ is guaranteed to be true. If $P \to Q$ is false, $\sim Q \to \sim P$ is guaranteed to be false.
• Proof Technique: To prove $P \to Q$, assume $\sim Q$ is true and show that it logically leads to $\sim P$.

Connected Concepts

• Converse
• Inverse
• Indirect Proof
• Necessary Condition
• Sufficient Condition
• Boolean Logic (Python)