One example is not a proof: how counterexamples test a mathematical rule

The words for every, for all, and always make a universal claim. The statement must survive every value in its stated domain. By contrast, a claim beginning with there exists only needs one successful example. These two logical shapes require different evidence.

Suppose someone claims that $a<b$ always implies $a^2<b^2$. Checking $2<3$ gives $4<9$, but this only confirms one case. The claim still promises something about negative numbers, zero, fractions, and every other allowed real value. Before calculating, write down the domain and the quantifier: exactly which values must the rule cover?

Choose $a=-3$ and $b=-2$. We have $-3<-2$, but squaring gives $9>4$. This single pair satisfies the premise and contradicts the conclusion, so the original universal claim is false. We do not need a second counterexample.

A counterexample must respect every condition in the claim. If a rule concerns positive numbers, a negative input cannot refute it. If it excludes zero, using zero is irrelevant. The useful habit is therefore precise: keep the assumptions, make the conclusion fail, and show both checks explicitly.

A good counterexample does more than say no. It often reveals the missing condition. Squaring preserves order when $0le a<b$, because both numbers are nonnegative. It also reverses the order of absolute sizes in some negative cases, which explains why the unrestricted statement failed.

The repaired claim is narrower but true: if $0le a<b$, then $a^2<b^2$. One proof factors the difference: $b^2-a^2=(b-a)(b+a).$ Under the assumptions, $b-a>0$ and $b+a>0$, so their product is positive. Hence $b^2-a^2>0$, which is equivalent to $a^2<b^2$.

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Counterexamples are rarely found by testing only comfortable whole numbers. Start with the edges of the domain: $0$, $1$, negative values, equal values, very small fractions, and values that make a denominator zero. If the claim uses a square root, logarithm, or reciprocal, check where that expression is defined.

For example, “adding the same number to the numerator and denominator leaves a fraction unchanged” may look plausible from the wording, but $1/2$ becomes $2/3$ after adding $1$ to both parts. The operation changes the ratio. Likewise, the claim $1/n<1$ for every positive integer $n$ fails at the boundary $n=1$, where equality holds. Boundary cases expose missing conditions efficiently.

A table, graph, or computer search can provide evidence and help discover a pattern. If the domain is finite and every case is checked, exhaustive verification can be a proof. But checking many cases from an infinite domain leaves untested values, however convincing the pattern looks.

This does not make examples unimportant. Examples help us understand the claim, catch arithmetic mistakes, and suggest the right proof. Counterexamples stress-test it. A proof then explains why the repaired statement must hold throughout its domain. Each tool answers a different question, and confusing their roles is the source of many fragile arguments.

Write the claim with its domain and quantifier. Test one ordinary case to confirm that you understand it. Then probe boundaries and sign changes, looking deliberately for a counterexample. If the claim survives, identify a structure that could prove it; if it fails, use the failure to revise the assumptions or conclusion. Finally, test the revised version again before writing the proof.

You can ask Euler’s tutor to challenge a conjecture with boundary cases, then request a proof only after the statement is precise. For more worked explanations with explicit checks, browse Euler Learn. The goal is not to distrust every pattern, but to know exactly what kind of evidence the pattern still needs.