Basics of Counterexamples and Disproving Universal Statements

A counterexample is a single specific instance that shows a universally quantified statement (a statement beginning with “for all”) is false. One failing case suffices to disprove universality. Therefore, the purpose of a counterexample is precisely to demonstrate that a universal statement is false by providing one situation in which the statement does not hold.

Universal statements are of the form “for all x, P(x).” To disprove such a claim, you must show that there exists at least one element x in the domain for which P(x) is false. A counterexample does exactly this: it is a particular instance that contradicts the claim of universality. Since universal statements require truth in all cases, showing even one failure is enough to negate the statement. In contrast, existential statements only need one case to be true, so counterexamples do not disprove them.

A counterexample never proves a statement is true; it instead shows that a universal statement is false. To prove something true, one must demonstrate that the statement holds in all relevant cases or derive it through valid logical argumentation. A counterexample does the opposite: by giving a single situation where the statement fails, it shows that the statement cannot be universally true. Therefore, the notion that a counterexample can prove truth is incorrect—counterexamples only prove falsehood of universal claims.

The statement “All primes are odd” asserts that every prime number has the property of being odd. The number 2 is a prime because it has exactly two positive divisors: 1 and itself. However, 2 is the only even prime number. Since it is prime yet not odd, it contradicts the universal claim. This makes 2 a valid counterexample that disproves the statement, because its existence shows that at least one prime is not odd, violating the universal claim.

The statement claims that every even number is divisible by 4. To disprove it, we must find an even number that is not divisible by 4. The number 6 is even because it is divisible by 2, but 6 divided by 4 does not yield an integer. Since an even number exists that fails to be divisible by 4, the universal statement is false. Therefore, 6 serves as an appropriate counterexample: it satisfies the hypothesis (being even) but not the asserted property (being divisible by 4).

Existential statements claim that there exists at least one element in the domain satisfying a property. To disprove an existential statement, you would need to show that no elements possess that property, which cannot be done with a single counterexample. Counterexamples are tools for disproving universal statements, which assert truth for all elements. Because existential claims require only one true instance, counterexamples—which show only one false instance—are irrelevant to disproving them.

The universal statement “All birds can fly” asserts that every bird possesses the ability to fly. Penguins are biologically classified as birds: they have feathers, lay eggs, and belong to the class Aves. However, penguins cannot fly; they are adapted for swimming instead. Because they fulfill the definition of a bird while failing the stated property, they contradict the universal claim. Thus, a penguin is a valid counterexample that disproves the statement.

To disprove the universal statement, we must find an x for which x² > x is false. If we let x = 0.5, then x² = 0.25. Since 0.25 is not greater than 0.5, the inequality fails. This shows that the claim does not hold for all real numbers, because at least one specific instance contradicts it. Therefore, x = 0.5 provides the required counterexample.

The statement claims that all integers are positive. To disprove this, we need a single integer that is not positive. The number −3 is an integer, but it is negative rather than positive. Because the existence of just one integer that fails to be positive contradicts the universal claim, −3 functions as a valid counterexample that negates the statement.

The statement says that all perfect squares must be even numbers. However, the number 9 is the square of 3, and 3 is odd. Thus 9 itself is odd, contradicting the claim that squares must always be even. Because 9 is indeed a perfect square but not an even number, it serves as a clear counterexample to the universal statement.

An existential statement requires at least one example where the property holds. To disprove an existential statement, one must show that no element satisfies the condition. A single counterexample only shows an instance where the property does not hold; it does not eliminate the possibility that another instance might satisfy it. Thus, counterexamples are insufficient for disproving existential statements and instead are used to disprove universal ones.

The statement claims that all mammals don’t lay eggs. Most mammals give live birth, but the platypus is unique: it is a mammal that lays eggs. For a universal claim, finding a single mammal that does not follow the stated property is enough to negate the statement. Because the platypus is a genuine mammal with the exceptional trait of egg-laying, it directly contradicts the claim, making it a valid counterexample.

We must find a positive number x for which x² 0 but does not satisfy x²

The statement asserts that every function is continuous. A step function is defined so that it jumps from one value to another at certain points, making it discontinuous. Because continuity fails at these jump points, the step function contradicts the claim that all functions are continuous. Thus, providing such a function disproves the universal statement by showing that at least one function lacks continuity.

The statement claims that every real number is greater than 0. To contradict this universal statement, we need a real number for which the inequality x > 0 is false. The negation of x > 0 is x ≤ 0. Therefore, any real number that is zero or negative will serve as a valid counterexample. Such an x disproves the original claim by providing a real number that does not satisfy the stated condition.