Counterexample Formula

The Formula

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

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

Quick Example

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

Notation

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

What This Formula Means

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

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

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

Worked Examples

Example 1

easy
Disprove: 'All prime numbers are odd.'

Solution

  1. 1
    To disprove a universal statement, find a single counterexample.
  2. 2
    Consider 2: it is prime (its only divisors are 1 and 2) and it is even.
  3. 3
    Since 2 is a prime number that is not odd, the statement is false.

Answer

\text{Counterexample: } 2 \text{ is prime and even.}
A counterexample is a specific case that shows a universal statement is false. Only one counterexample is needed to disprove a 'for all' claim.

Example 2

medium
Disprove: 'For all real numbers x, x^2 > x.'

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

Why This Formula Matters

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

Frequently Asked Questions

What is the Counterexample formula?

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

How do you use the Counterexample formula?

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

What do the symbols mean in the Counterexample formula?

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

Why is the Counterexample formula important in Math?

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

What do students 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 the Counterexample formula?

Before studying the Counterexample formula, you should understand: quantifiers.

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