Counterexamples in Number Theory, Geometry, and Functions

The statement claims that all triangles have equal sides, meaning every triangle would have to be equilateral. A scalene triangle is defined to have all three sides of different lengths. Because a scalene triangle is still a triangle yet violates the claimed property of having equal sides, it directly contradicts the universal statement. The existence of even one scalene triangle is enough to prove that not all triangles possess equal side lengths.

The statement asserts that every even number must be positive. The number −2 is an even integer because it is divisible by 2, but it is not positive. This means the statement fails for at least one value in its domain. Because universal statements require truth for all cases, the single example of −2 is sufficient to falsify the claim and serves as a valid counterexample.

A counterexample demonstrates that a universal statement is false by providing one instance in which it fails. Existential statements, however, require showing that at least one satisfying example exists. Because counterexamples aim to show failure of a universal claim and do not establish that something exists, they cannot prove existential statements. They are tools for disproving, not proving, and thus cannot verify existential claims.

The statement says all polynomials are linear, meaning every polynomial would have to be of degree 1. The function f(x) = x² is a polynomial, but it has degree 2, which violates the claimed property of being linear. Because this function is still a genuine polynomial yet does not satisfy the condition stated, it disproves the universal statement. A single polynomial of degree higher than 1 is all that is needed to contradict the claim.

The statement asserts that every real number is rational. A rational number can be expressed as a ratio of two integers. π cannot be expressed as a ratio of integers; it is an irrational real number. Because π is a real number that does not satisfy the property of being rational, it directly contradicts the universal claim. Thus, π serves as a valid counterexample proving the statement false.

Direct proofs attempt to show that a statement is true by deriving it logically from assumptions. Counterexamples do the opposite; they aim to disprove universal statements. Since counterexamples do not demonstrate truth but instead provide a contradiction to the claim, they are not part of direct proofs. They serve as a method of disproof rather than proof, and therefore are not used in direct proofs.

The statement claims that every odd number is prime. The number 9 is odd, but it is not prime because it has divisors other than 1 and itself (specifically 3). Since 9 satisfies the hypothesis of being odd yet fails the property of being prime, it contradicts the universal statement. Therefore, 9 is a valid counterexample showing that not all odd numbers are prime.

The statement claims that every pair of lines is parallel. To contradict this, you need an example of lines that are not parallel. Intersecting lines meet at a point, which means they are not parallel. Their existence disproves the idea that all lines share the property of never meeting. Thus, a pair of intersecting lines provides a clear counterexample to the universal statement.

Counterexamples work by finding a single violating case; the size of the domain does not matter. Even in a finite domain, showing that one example contradicts the statement is enough to refute the universal claim. Whether the domain contains three elements or infinitely many, only one failure is needed. Thus, counterexamples do not depend on the domain being infinite.

The statement asserts that every number must be a perfect square. A perfect square is a number that can be written as n² for some integer n. The number 3 cannot be written as an integer square. Because 3 is a number that fails to satisfy the condition of being a perfect square, it disproves the universal statement. Therefore, 3 is a valid counterexample.

The statement claims that every infinite series converges. The harmonic series, defined by the sum of 1/n, is a classical example of a divergent series. Even though its terms approach zero, the sum grows without bound. Because the harmonic series is a legitimate infinite series that does not converge, it provides a counterexample that disproves the universal statement.

A conditional statement “If P, then Q” is false only in the case where P is true but Q is false. In all other cases, the conditional is true. Therefore, to disprove a conditional statement, the counterexample must satisfy the hypothesis but violate the conclusion. This creates the precise situation in which the implication fails, making it the correct form of a counterexample.

Prime numbers are defined as numbers with exactly two positive divisors: 1 and themselves. The number 2 meets this definition and is therefore prime. However, 2 is even. Because the statement claims that all prime numbers are odd, the existence of the even prime number 2 contradicts it. Thus, 2 is a valid counterexample.

To disprove the universal statement, we must find an integer n for which n² > n is false. If n = 1, then n² = 1, which is not greater than 1. Since the inequality does not hold for n = 1, it contradicts the universal claim. Therefore, n = 1 is a valid counterexample because it is an integer that fails to satisfy the required property.

The statement claims that if x² = 25, then x must equal 5. However, both 5 and −5 square to 25. The value x = −5 satisfies the hypothesis but does not satisfy the conclusion. This produces the exact situation required for a counterexample to a conditional: the hypothesis is true while the conclusion is false. Therefore, −5 is the correct counterexample.