1. Idea of the Contrapositive
The contrapositive of a statement is formed by reversing and negating both the hypothesis and the conclusion. It takes the original "If P, then Q" and produces "If not Q, then not P".
The contrapositive keeps the logical meaning of the original implication, even though it changes the order and negates both parts.
1.1. Why the contrapositive matters
In proofs and reasoning, the contrapositive is often easier to work with than the original statement. Showing that "If not Q, then not P" is true automatically proves "If P, then Q".
2. Formal Definition
If the original implication is:
\(P \Rightarrow Q\)
then the contrapositive is:
\(\neg Q \Rightarrow \neg P\)
where both parts are negated and their positions are swapped.
2.1. Understanding the structure
The contrapositive reverses the direction of implication and applies negation. It checks whether the failure of Q forces the failure of P.
3. Relation to the Original Statement
The contrapositive is logically equivalent to the original statement. This means they always have the same truth value.
If one is true, the other must also be true; if one is false, the other must also be false.
3.1. Why the equivalence holds
Both statements fail only in the same situation: when P is true and Q is false. Because of this precise match, proving the contrapositive automatically proves the original implication.
4. Examples
These examples show how a contrapositive is formed and why it is often more convenient to verify.
4.1. Everyday example
Original: "If it rains, then the ground becomes wet."
Contrapositive: "If the ground does not become wet, then it did not rain."
This is logically equivalent to the original statement.
4.2. Mathematical example
Original: "If a number is a multiple of 4, then it is even."
Contrapositive: "If a number is not even, then it is not a multiple of 4."
Both forms are true.
5. Notes and Observations
Important points about the contrapositive:
- It reverses and negates both parts of the original statement.
- It is logically equivalent to the original implication.
- Contrapositive proofs are widely used when direct proofs are difficult.
- The contrapositive is always true exactly when the original implication is true.