Math 2.0

General and overarching topics Concept

Counterexample

Builds on Universal quantifierNegation of a universal statement

Summary

A counterexample to a universal statement ∀x∈X, P(x)\forall x \in X, \, P(x) is an element a∈Xa \in X such that P(a)P(a) is false. One counterexample shows that the universal statement is false.

Remarks

The number 22 is a counterexample to "each prime number is odd". This works because of the negation of a universal statement: ¬∀x P(x)\neg \forall x \, P(x) is equivalent to ∃x ¬P(x)\exists x \, \neg P(x).

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Proposition, concept, no evidence levelPropositionconceptNegation, definition, no evidence levelNegationdefinitionPredicate, definition, no evidence levelPredicatedefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionNegation of a universal statement, statement, no evidence levelNegation of a univers…statementCounterexample, concept, no evidence levelCounterexampleconcept
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Proposition concept
  2. Negation definition
  3. Predicate definition
  4. Existential quantifier definition
  5. Universal quantifier definition
  6. Negation of a universal statement statement
  7. Counterexample concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Counterexample, lead section
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Tag0ED72CTBNP
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Kindconcept
WritingW0 stub
MSC00A35
UsesUniversal quantifier, Negation of a universal statement
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