Counterexample

Logic
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Also known as: counter-example, disproof by example

Grade 9-12

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A counterexample is a specific instance that satisfies the hypothesis of a claim but contradicts its conclusion, thereby disproving the general statement. One counterexample instantly kills any universal claim — it is the most efficient form of mathematical disproof.

Definition

A counterexample is a specific instance that satisfies the hypothesis of a claim but contradicts its conclusion, thereby disproving the general statement.

💡 Intuition

One case where it fails is enough to kill a 'for all' claim.

🎯 Core Idea

To disprove \forall x\, P(x), find one x where P(x) is false.

Example

'All primes are odd' — counterexample: 2. 'x^2 > x' — counterexample: x = 0.5.

Formula

\neg(\forall x\, P(x)) \Leftrightarrow \exists x\, \neg P(x)

Notation

To disprove \forall x\, P(x), exhibit a specific x_0 such that \neg P(x_0)

🌟 Why It Matters

One counterexample instantly kills any universal claim — it is the most efficient form of mathematical disproof.

💭 Hint When Stuck

Try small, simple values first (0, 1, 2, -1, 1/2). Counterexamples are usually lurking among the simplest cases.

Formal View

\neg(\forall x\,P(x)) \Leftrightarrow \exists x_0\,\neg P(x_0); a single witness x_0 with \neg P(x_0) refutes the universal claim

🚧 Common Stuck Point

A counterexample disproves "for all" claims, but finding many examples that work does NOT prove a universal statement is true.

⚠️ Common Mistakes

  • Trying to use a counterexample to prove a statement true — counterexamples can only disprove universal claims
  • Finding one example that works and concluding the statement is always true — one positive example does not prove \forall x\, P(x)
  • Giving a counterexample that does not actually satisfy the hypothesis — e.g., 'disproving' a claim about primes by testing a composite number

Frequently Asked Questions

What is Counterexample in Math?

A counterexample is a specific instance that satisfies the hypothesis of a claim but contradicts its conclusion, thereby disproving the general statement.

Why is Counterexample important?

One counterexample instantly kills any universal claim — it is the most efficient form of mathematical disproof.

What do students usually get wrong about Counterexample?

A counterexample disproves "for all" claims, but finding many examples that work does NOT prove a universal statement is true.

What should I learn before Counterexample?

Before studying Counterexample, you should understand: quantifiers.

Prerequisites

quantifiers

Next Steps

proof intuition

How Counterexample Connects to Other Ideas

To understand counterexample, you should first be comfortable with quantifiers. Once you have a solid grasp of counterexample, you can move on to proof intuition.

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