Counterexample Formula

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

The Formula

¬(∀x P(x))⇔∃x ¬P(x)\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: 22. 'x2>xx^2 > x' — counterexample: x=0.5x = 0.5.

Notation

To disprove ∀x P(x)\forall x\, P(x), exhibit a specific x0x_0 such that ¬P(x0)\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

¬(∀x P(x))⇔∃x0 ¬P(x0)\neg(\forall x\,P(x)) \Leftrightarrow \exists x_0\,\neg P(x_0); a single witness x0x_0 with ¬P(x0)\neg P(x_0) refutes the universal claim

Worked Examples

Example 1

easy
Disprove: 'All prime numbers are odd.'

Answer

Counterexample: 2 is prime and even.\text{Counterexample: } 2 \text{ is prime and even.}

First step

1
To disprove a universal statement, find a single counterexample.

Full solution

  1. 2
    Consider 22: it is prime (its only divisors are 1 and 2) and it is even.
  2. 3
    Since 22 is a prime number that is not odd, the statement is false.
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 xx, x2>xx^2 > x.'

Example 3

medium
Find a counterexample to 'sin⁡(x+y)=sin⁡x+sin⁡y\sin(x + y) = \sin x + \sin y for all real x,yx, y.'

Common Mistakes

  • Giving a confirming example and thinking it proves the claim - one supporting case cannot establish a universal statement.
  • Producing a case that fails the hypothesis - a valid counterexample must satisfy the 'if' part and break the 'then' part.
  • Trying to disprove a claim that was only about 'some' - a counterexample refutes 'for all', not an existence claim.

Why This Formula Matters

Disproving and proving are asymmetric: a universal claim needs a general proof to confirm but only one counterexample to destroy. Knowing this saves enormous effort — instead of attempting a doomed proof, you hunt the single case that breaks the claim, which is also how mathematicians sharpen conjectures. Recognizing it by "Am I trying to kill a universal claim by exhibiting one case that fits the hypothesis but breaks the conclusion?" — rather than by familiar numbers — is what lets a student tell it apart from proof and edge case and confirming example in a mixed problem set.

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 ∀x P(x)\forall x\, P(x), exhibit a specific x0x_0 such that ¬P(x0)\neg P(x_0)

Why is the Counterexample formula important in Math?

Disproving and proving are asymmetric: a universal claim needs a general proof to confirm but only one counterexample to destroy. Knowing this saves enormous effort — instead of attempting a doomed proof, you hunt the single case that breaks the claim, which is also how mathematicians sharpen conjectures. Recognizing it by "Am I trying to kill a universal claim by exhibiting one case that fits the hypothesis but breaks the conclusion?" — rather than by familiar numbers — is what lets a student tell it apart from proof and edge case and confirming example in a mixed problem set.

What do students get wrong about Counterexample?

The procedure for counterexample is the easy part; the trap is giving a confirming example and thinking it proves the claim. Asking "Am I trying to kill a universal claim by exhibiting one case that fits the hypothesis but breaks the conclusion?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Counterexample formula?

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

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