TeXipedia

exist

Denotes logical existence, indicating that there is at least one element satisfying a given condition.

Overview

Essential in mathematical logic, set theory, and formal proofs where existence claims need to be expressed rigorously.

  • Commonly used in theorem statements and mathematical definitions
  • Appears frequently in abstract algebra and analysis to specify properties of elements
  • Often paired with quantifiers and logical connectives in formal mathematical writing
  • Particularly important in constructive mathematics and proof writing

Examples

Expressing the existence of a solution to an equation

∃x∈R:x2+2x+1=0\exists x \in \mathbb{R} : x^2 + 2x + 1 = 0 
\exists x \in \mathbb{R} : x^2 + 2x + 1 = 0

Stating the existence of a unique element in set theory

∃!x∈A:f(x)=0\exists! x \in A : f(x) = 0 
\exists! x \in A : f(x) = 0

Negating existence in a logical statement

¬∃x∈N:x2<0\neg \exists x \in \mathbb{N} : x^2 < 0 
\neg \exists x \in \mathbb{N} : x^2 < 0