If-Then Statements: Examples, Converse and Counterexamples
An if-then conditional has a hypothesis and a conclusion. In “If a quadrilateral is a square, then it is a rectangle,” being a square is the hypothesis and being a rectangle is the conclusion. To disprove a universal conditional, find one case where the hypothesis is true and the conclusion is false.
Find the hypothesis and conclusion
The form is \(p\rightarrow q\): if \(p\), then \(q\). Read the condition after “if” as the hypothesis and the result after “then” as the conclusion. The statement asserts a one-way implication; it does not automatically assert the reverse.
Example 1: “If it is raining, then the ground is wet.” Here \(p\) is “It is raining” and \(q\) is “The ground is wet.” This illustrates the sentence structure; a real-world claim needs a stated setting, because sheltered ground need not get wet.
When “if” comes second: “A quadrilateral is not a rectangle if it has no right angles” means “If a quadrilateral has no right angles, then it is not a rectangle.” The hypothesis is having no right angles. The conclusion is not being a rectangle. Use the word “if,” rather than sentence order, to identify the hypothesis.
Compare the four forms
Use the same universe throughout: ordinary plane quadrilaterals. Let \(p\) mean “the quadrilateral is a square” and \(q\) mean “the quadrilateral is a rectangle.”
| Form | Rule | Example |
|---|---|---|
| Conditional | \(p\rightarrow q\) | If it is a square, then it is a rectangle. |
| Converse | \(q\rightarrow p\) | If it is a rectangle, then it is a square. |
| Inverse | \(\neg p\rightarrow\neg q\) | If it is not a square, then it is not a rectangle. |
| Contrapositive | \(\neg q\rightarrow\neg p\) | If it is not a rectangle, then it is not a square. |
The original and its contrapositive are logically equivalent. The converse and inverse are equivalent to each other, but neither is generally equivalent to the original. In this example, the original and contrapositive are true. A 2-by-3 rectangle makes both the converse and inverse false.
Example 2: For integers, “If a number is even, then it is divisible by 2” has a true converse too, because divisibility by 2 defines evenness. This is a case where all four forms are true. Do not use that special case to conclude that every true conditional has a true converse.
How to find a counterexample
For a universal if-then statement, choose an object that satisfies the hypothesis, then test the conclusion. A case with a false hypothesis does not disprove the statement.
- “If an animal is a mammal, then it is a cat.” False: a dog is a mammal and is not a cat.
- “If an animal is a bird, then it can fly.” False: a penguin is a bird and cannot fly.
- “If a shape has four sides, then it is a rectangle.” False: a nonrectangular rhombus has four sides but does not have four right angles. A square is a rectangle, so a square is not a counterexample.
For the cat statement, the converse is “If an animal is a cat, then it is a mammal,” which is true. That does not rescue the original: test each direction independently.
Truth values and “if and only if”
In classical propositional logic, a material conditional is false only in the true-hypothesis, false-conclusion case. The table evaluates one assignment of truth values at a time. A universal mathematical claim must hold for every object in its stated domain.
| \(p\) | \(q\) | \(p\rightarrow q\) |
|---|---|---|
| True | True | True |
| True | False | False |
| False | True | True |
| False | False | True |
A biconditional, \(p\leftrightarrow q\), says “p if and only if q.” It requires both directions. For example, an integer is even if and only if it is divisible by 2. Being a square and being a rectangle are not equivalent conditions.
Practice with answers
- Write the converse, inverse and contrapositive of “If a bird is a penguin, then it cannot fly.” Keep the domain restricted to birds.
- Is “If a plane polygon has three sides, then it is a triangle” true?
- For “If an animal is a cat, then it is a mammal,” identify the converse: (a) If it is a mammal, then it is a cat; (b) If it is not a cat, then it is not a mammal; (c) If it is not a mammal, then it is not a cat.
Show answers and reasoning
- Converse: If a bird cannot fly, then it is a penguin. Inverse: If a bird is not a penguin, then it can fly. Contrapositive: If a bird can fly, then it is not a penguin. An ostrich disproves the converse and inverse; the contrapositive has the same truth value as the original.
- True, by the definition of a triangle as a three-sided plane polygon.
- Choice (a), because it swaps the hypothesis and conclusion. Choice (b) is the inverse; choice (c) is the contrapositive. The converse is false: a dog provides a counterexample.
Further review
For a reference on logical equivalence and the four forms, see OpenStax Contemporary Mathematics: Equivalent Statements. In a geometry proof, record the given conditions before choosing a theorem, and check that the theorem applies in the direction you need.
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