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.

What is the Counterexample formula?

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

When do you use Counterexample?

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

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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