TeXipedia

nexists

Denotes the logical concept of non-existence, indicating that no element or value satisfies a given condition.

Overview

Essential in mathematical logic, set theory, and formal proofs where negating existence claims is necessary.

  • Commonly used in advanced mathematics to express that certain elements or solutions do not exist
  • Appears frequently in abstract algebra and topology to prove impossibility results
  • Often paired with quantifiers and set-builder notation to construct precise mathematical statements
  • Particularly useful in contradiction proofs and counterexamples

Examples

Stating that there is no solution to an equation.

∄x∈R:x2<0\nexists x \in \mathbb{R} : x^2 < 0 
\nexists x \in \mathbb{R} : x^2 < 0

Expressing that a specific element does not exist in a set.

∄n∈N:n2=2\nexists n \in \mathbb{N} : n^2 = 2 
\nexists n \in \mathbb{N} : n^2 = 2

Showing non-existence in a logical statement.

∄x∈A:P(x)∧¬P(x)\nexists x \in A : P(x) \land \neg P(x) 
\nexists x \in A : P(x) \land \neg P(x)