Examples, Nonexamples, and Counterexamples

The Big Idea

Definitions tell us what a mathematical idea means, but a definition can feel flat until we see it at work. Examples give the definition something to hold. Nonexamples reveal where it stops. Counterexamples test whether a broad claim has reached too far.

These three tools perform different jobs:

• An example satisfies the definition or condition being discussed.
• A nonexample does not satisfy the definition or condition.
• A counterexample is an allowed case that makes a general claim false.

The easiest way to remember the difference is this: examples and nonexamples help us understand a category; counterexamples challenge a claim.

A Quick Comparison

Examples Bring Definitions to Life

An example is an object or case that satisfies the relevant definition. Suppose an even integer is defined as an integer that can be written as \(2k\) for some integer \(k\).

The integer \(14\) is an example because

\[ 14=2(7). \]

The integer \(-8\) is also an example because

\[ -8=2(-4). \]

The second example is especially useful because it prevents us from quietly assuming that even numbers must be positive. Good examples do more than confirm the first picture that came to mind; they show the range allowed by the definition.

One Example Is Rarely Enough for Understanding

If every square shown in a lesson is upright, equal in size, and centered on the page, a student may accidentally learn “square-looking picture” instead of the definition. A tilted square, a very small square, and a square inside another figure help separate essential properties from accidental appearance.

A useful collection of examples varies features that do not affect membership:

• Size
• Position
• Orientation
• Notation
• Sign
• Units
• Order of presentation
• Level of complexity

What should remain fixed are the defining conditions. When irrelevant details change but the object remains an example, the definition becomes more visible.

An Example Can Prove Existence

A single example cannot usually prove that a property holds for every object. It can, however, prove that at least one such object exists.

Consider the statement:

There exists an even prime number.

The number \(2\) proves the statement. It is even, and its only positive divisors are \(1\) and \(2\).

This works because an existential claim asks for at least one successful case. A universal claim asks for every case, which requires a different kind of justification.

Nonexamples Reveal the Boundary

A nonexample is an object that does not satisfy the definition being studied. A useful nonexample does not merely sit far away from the category. It helps us identify which required condition is missing.

Suppose a square is defined as a quadrilateral with four equal sides and four right angles.

• A \(3\times5\) rectangle is a nonexample because its four sides are not all equal.
• A non-square rhombus is a nonexample because its angles are not all right angles.
• An equilateral triangle is a nonexample because it has three sides, not four.
• An open figure made of four equal segments is a nonexample because it is not a closed quadrilateral.

Each case fails for a different reason. Together, they teach the complete definition more effectively than one distant nonexample would.

Near-Miss Nonexamples Are Especially Valuable

A near-miss nonexample satisfies many parts of a definition but fails one important condition. Because it comes close, it forces us to pay attention to the exact boundary.

For the definition “a prime number is a positive integer greater than 1 with exactly two positive divisors”:

• \(5\) is an example.
• \(6\) is a nonexample because it has more than two positive divisors.
• \(1\) is a near-miss nonexample because it is positive but has only one positive divisor.
• \(-5\) is a nonexample under this definition because it is not a positive integer greater than 1.
• \(\frac52\) is a nonexample because it is not an integer.

The cases \(1\) and \(-5\) are useful because they expose conditions that students often omit when casually remembering the definition.

Counterexamples Disprove General Claims

A counterexample is a specific case that satisfies the starting conditions of a general claim but makes its conclusion false.

Consider the claim:

Every prime number is odd.

The number \(2\) is a counterexample. It satisfies the starting condition—it is prime—but it fails the conclusion because it is even.

One counterexample is enough because the word every allows no exceptions. The claim may describe nearly all primes, but “nearly all” is not what it said.

The Anatomy of a Counterexample

Many universal claims have the form:

\[ \text{For every }x,\quad \text{if }P(x)\text{ is true, then }Q(x)\text{ is true.} \]

A genuine counterexample must do three things:

1. Belong to the stated domain.
2. Make the hypothesis \(P(x)\) true.
3. Make the conclusion \(Q(x)\) false.

To disprove “Every rectangle is a square,” a \(3\times5\) rectangle works. It is a rectangle, so it satisfies the hypothesis. It is not a square, so it violates the conclusion.

A triangle would not work as a counterexample. It is not a rectangle, so the original claim never promised anything about it.

The Same Object Can Perform Different Jobs

A \(3\times5\) rectangle is:

• An example of a rectangle
• A nonexample of a square
• A counterexample to the claim “Every rectangle is a square”

The object did not change. The question changed. “Example,” “nonexample,” and “counterexample” describe how a case is being used relative to a definition or claim.

Many Examples Still Do Not Prove “Every”

Suppose the first few calculations give prime numbers:

\[ 1^2+1+1=3,\qquad 2^2+2+1=7,\qquad 3^2+3+1=13. \]

These examples may tempt us to claim that \(n^2+n+1\) is prime for every positive integer \(n\). The next case gives

\[ 4^2+4+1=21, \]

which is composite. The first three successful examples suggested a conjecture; the fourth case disproved it. Testing cases is excellent for forming conjectures and finding errors, but proof is needed to justify a universal conclusion.

Even a computer checking millions of cases has checked only those cases unless the domain itself is finite and completely exhausted. A universal claim over infinitely many objects cannot be established by a finite list of successful examples alone.

Look at the Edges

False generalizations often survive ordinary examples and fail at boundary or exceptional cases. When testing a definition or conjecture, remember to try:

• Zero and one
• Negative numbers
• Fractions and irrational numbers
• The smallest or largest permitted values
• Equal values where distinct values were expected
• Repeated entries
• Empty sets or empty lists
• Endpoints of intervals
• Vertical, horizontal, or degenerate geometric cases
• Values that make a denominator zero or a radical undefined

These cases are not tricks. They are legitimate members—or deliberately excluded nonmembers—whose behavior reveals whether the statement was written carefully.

How to Search for a Counterexample

Suppose someone claims, “The square of every real number is greater than the number.” Rather than testing random large values, examine the relationship

\[ x^2\gt x. \]

Move everything to one side:

\[ x^2-x=x(x-1). \]

Between \(0\) and \(1\), one factor is positive and the other is negative, so \(x(x-1)\lt0\). Choosing \(x=\frac12\) gives

\[ \left(\frac12\right)^2=\frac14\not\gt\frac12. \]

The value \(\frac12\) is a counterexample. Structure guided the search more efficiently than guessing.

What Does Not Count as a Counterexample?

A Counterexample Can Help Repair a Claim

Finding a counterexample is not the end of mathematical thinking. Ask why the claim failed and whether a more precise statement captures the intended pattern.

The false claim

Every prime number is odd

can be repaired as

Every prime number greater than 2 is odd.

The counterexample \(2\) revealed the missing condition.

Sometimes the repaired statement restricts the domain. Sometimes it weakens the conclusion from “every” to “some,” “often,” or “under these assumptions.” A good counterexample tells us exactly where the original claim broke.

Examples, Conjectures, Counterexamples, and Proof

These tools often work in a productive cycle:

1. Study examples and notice a pattern.
2. State the pattern as a conjecture.
3. Test ordinary and boundary cases.
4. If a counterexample appears, revise or reject the conjecture.
5. If no counterexample appears, seek a proof.
6. Use the proof to explain why every allowed case works.

Counterexample search and proof are not competitors. Both are methods for testing whether reasoning has earned a general conclusion.

Examples Help Classification

When learning a new category, build a small test collection:

1. Choose a simple example.
2. Choose an unusual but valid example.
3. Choose a near-miss nonexample for each defining condition.
4. Explain why every case belongs or does not belong.
5. Try to formulate a false overgeneralization and find a counterexample to it.

For a function defined as a relation assigning exactly one output to each input, include an ordinary numerical function, a constant function, and a function represented by a table. Then include a relation where one input has two outputs as a near-miss nonexample.

The goal is not to collect random cases. It is to map the boundary created by the definition.

A Friendly Case-Checking Routine

When you are given a definition or claim, ask:

1. What is the exact definition or statement? Do not rely on a vague memory.
2. What is the domain? Which objects are eligible?
3. Which conditions are required? Separate the hypothesis from the conclusion.
4. Can I produce a simple example?
5. Can I produce an unusual example?
6. Can I change one required condition to make a near-miss nonexample?
7. Does the claim use “every,” “all,” or “always”? Search actively for a counterexample.
8. Does my proposed counterexample satisfy the hypothesis and violate the conclusion?
9. If the claim fails, how might it be repaired?

Working Summary

An example satisfies a definition or condition. A nonexample does not. A counterexample satisfies a claim's starting conditions but violates its conclusion, thereby disproving the claim.

Examples develop intuition and can prove existence. Nonexamples sharpen definitions. Counterexamples prevent overgeneralization and help repair false conjectures. Together, they turn mathematical classification and reasoning into active investigation.

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