Negation of a universal proposition

Statement

Write the negation of “all equilateral triangles have three axes of symmetry”.

Solution

The proposition has the form $\forall x\,P(x)$. Its negation is $\exists x\,\lnot P(x)$: the universal quantifier becomes existential and the inner property is negated.

In English: “there exists (at least) one equilateral triangle that does not have three axes of symmetry”.

(The negation is not “no equilateral triangle has three axes of symmetry”: a single counterexample suffices to negate a universal statement.)

Links

Topics: Set theory
Concepts: Quantifiers
Skills: Reasoning by cases
Exercise type: Word problem