Converse, Inverse, and Contrapositive
Every conditional statement has three related forms. One of them is always logically equivalent to the original — do you know which?
Introduction
Given any “if p, then q” statement, we can form three new statements by swapping or negating the hypothesis and conclusion. Understanding these forms is essential for proof writing.
Past Knowledge
Conditional statements & truth values (2.1.3).
Today's Goal
Write the converse, inverse, and contrapositive and determine their truth values.
Future Success
Biconditionals (2.1.5), indirect proofs, and the contrapositive proof technique.
Key Concepts
| Form | Symbolic | How to Build It |
|---|---|---|
| Conditional | Original | |
| Converse | Swap hypothesis & conclusion | |
| Inverse | Negate both | |
| Contrapositive | Swap and negate both |
The Golden Rule
A conditional and its contrapositive always have the same truth value. The converse and inverse also share the same truth value as each other (but not necessarily with the original).
Worked Examples
Writing All Four Forms
Conditional: “If it rains, then the ground is wet.”
Converse: If the ground is wet, then it rains.
Inverse: If it does not rain, then the ground is not wet.
Contrapositive: If the ground is not wet, then it did not rain.
The conditional and contrapositive are true. The converse and inverse are false (sprinklers could wet the ground).
Geometric Conditional
“If two angles are vertical angles, then they are congruent.” Write the contrapositive.
Contrapositive: If two angles are not congruent, then they are not vertical angles.
Both the original and contrapositive are true.
Common Pitfalls
Assuming the Converse Is True
Just because a conditional is true does not mean its converse is true. “If it's a dog, then it's an animal” is true, but “If it's an animal, then it's a dog” is false.
Real-Life Applications
Medical Diagnosis
“If a patient has the flu, then they have a fever.” The converse — “If they have a fever, then they have the flu” — is clearly false (many conditions cause a fever). Doctors use the contrapositive: “If no fever, then no flu” — which is logically guaranteed.
Practice Quiz
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