SECONDARY MATHEMATICS · PROOF FOUNDATIONS
To prove “all”, examples are not enough. To disprove “all”, one valid exception can be decisive.
Read the quantifier before calculating
A universal claim says that every permitted object has a property: every integer of a certain kind, every triangle satisfying specified conditions, or every real number in a stated domain. To disprove such a claim, find one object that satisfies the conditions but fails the conclusion.
That object is a counterexample. The logical force comes from belonging to the claimed set and violating the claimed property.
Example 1: “Every prime is odd”
The number 2 is prime and even. Therefore the statement “every prime number is odd” is false. Listing 3, 5, 7, 11 and 13 does not rescue the universal claim. One permitted exception is enough.
Example 2: “Squaring makes a number larger”
If the claim is “for every positive real x, x² > x”, choose x = 1/2. Then x² = 1/4, which is smaller than 1/2. The claim is false.
This counterexample also suggests a better conjecture. For x > 1, x² > x; for 0 < x < 1, x² < x; and at x = 1 they are equal. Disproof can improve a statement rather than merely end the investigation.
Example 3: geometry
Claim: “Every quadrilateral with four equal sides is a square.” A non-square rhombus has four equal sides but does not have four right angles. It is therefore a counterexample.
Notice that an arbitrary sketch is not enough. The counterexample must genuinely satisfy the premise. A quadrilateral with unequal sides would not test this claim because it does not belong to the stated class.
Example 4: algebraic cancellation
Claim: “For every real x, (x² − 4)/(x − 2) = x + 2.” At x = 2, the left side is undefined because its denominator is zero, while the right side equals 4. Therefore the statement as written for every real x is false.
A corrected statement is: for every real x with x ≠ 2, (x² − 4)/(x − 2) = x + 2. The restriction repairs the claim.
A counterexample must obey every condition
If the claim says “every positive integer divisible by 6 is even”, the number 9 cannot disprove it because 9 is not divisible by 6. A proposed counterexample outside the domain has no logical force against the statement.
This is why domain reading is part of reasoning. Before celebrating an exception, check that it satisfies all premises.
Search strategically
- Try boundary values such as 0 or 1 when permitted.
- Try negative numbers if the domain includes them.
- Try fractions between 0 and 1 when powers or reciprocals appear.
- In geometry, test special subclasses: square, rectangle, rhombus, isosceles or right triangle where relevant.
- Check excluded values created by denominators, square roots or other restrictions.
Strategic testing is not random guessing. It targets places where the proposed rule is most likely to change behaviour.
Counterexample versus contradiction
If the target is a universal claim and a concrete exception is easy to produce, counterexample is usually the cleanest disproof. Proof by contradiction serves a different role: assume the genuine negation of a statement and show that the assumption cannot be consistent with the established conditions.
Do not turn a one-line counterexample into a page of unnecessary algebra. Mathematical strength includes choosing an efficient valid argument.
Counterexamples cannot prove a universal claim
If you test 100 cases and all work, you have not produced a counterexample—but you have also not automatically proved the universal statement. The asymmetry matters: one failing case can disprove “all”, while many successful cases may only support a conjecture.
Move from examples to conjecture, then to an appropriate proof when the statement survives testing.
Worked disproof set
Claim A: Every multiple of 4 is a multiple of 8. Counterexample: 4.
Claim B: For every real x, x² ≥ x. Counterexample: x = 1/2, since 1/4 < 1/2.
Claim C: Every rectangle has four equal sides. Counterexample: a 2-by-5 rectangle.
Claim D: If two numbers have the same square, then they are equal. Counterexample: 3 and −3 have equal squares but are not equal.
Repair the false claim
After finding a counterexample, ask what condition would make the statement true. “Every multiple of 4 is a multiple of 8” can be replaced by “every multiple of 8 is a multiple of 4”. “For every real x, x² ≥ x” can be restricted to x ≤ 0 or x ≥ 1.
This repair step connects disproof to necessary and sufficient conditions. The goal is not only to catch false statements but to understand exactly what makes a true statement work.
Practice
- Disprove: every odd number is prime.
- Disprove: for all real x, |x| = x.
- Disprove: every parallelogram is a rectangle.
- Disprove: if ab = 0, then both a and b equal zero.
Possible answers: 9 is odd but composite; x = −2 gives |x| = 2 ≠ −2; a non-rectangular parallelogram; a = 0 and b = 5 gives ab = 0 although b ≠ 0.
Continue
Read What Counts as Proof?, Direct Proof, and Proof by Contradiction. The broader article Counterexamples in Secondary Mathematics develops counterexample thinking across the subject. Return to the Secondary Mathematics Master Index.