There exists (∃) in LaTeX
\exists typesets the existential quantifier ∃, read 'there exists', used to assert that at least one object in a domain satisfies a given property.
LaTeX command
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The existential quantifier claims that some object with a stated property is out there, without saying how many or how to find it. \exists x : x^2 = 2 asserts that a real number squaring to 2 exists, true because of \sqrt{2}, but the statement itself proves nothing about uniqueness.
If exactly one object should satisfy the property, use the unique-existence quantifier \exists! instead of plain \exists. \exists! x, P(x) is shorthand for 'there exists an x with property P, and it is the only one', which otherwise takes two separate clauses to state in full.
\exists is negated with \nexists ('there is no', needs amssymb, available by default in KaTeX), or equivalently by writing \neg \exists and pushing the negation inward to become \forall x, \neg P(x). Both are common, but \nexists is the more compact and readable choice at the start of a statement.
Other forms
\nexiststhere is no (∄)Examples
$\exists x : x^2 = 2$there exists a real number whose square is 2
$\exists! x \in \mathbb{R},\ x^3 = 8$unique existence, exactly one such x
$\nexists x \in \mathbb{Q},\ x^2 = 2$no rational number squares to 2
$\exists \delta > 0 \text{ such that } |x-a| < \delta$existence with a plain-language 'such that'
Tips and common mistakes
- Separate the quantified variable from the property with a colon or the words 'such that', either
\existsx : P(x) or\existsx\text{ such that }P(x), whichever reads more naturally for your audience. - Use
\exists! for unique existence rather than adding a separate uniqueness sentence afterwards; it is standard notation in analysis and algebra. \nexistsneeds the amssymb package in a full LaTeX document (KaTeX on this site supports it by default); if it fails to render elsewhere, fall back to\neg \exists.- An existence statement is not constructive by itself:
\existsx, P(x) says such an x is out there, but proving it may still require exhibiting one explicitly or arguing indirectly. - Watch quantifier order when
\existsappears with\forall, swapping\existsy\,\forallx for\forallx\,\existsy changes the meaning of the statement.
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What does \exists mean?
It is the existential quantifier, read 'there exists'. \exists x, P(x) claims that at least one value of x in the domain makes P(x) true, without saying how many or identifying one.
What is the symbol for 'there exists a unique'?
\exists!, typeset as an exclamation mark after the existential quantifier, meaning exactly one object with the stated property exists.
How do I write 'there does not exist' in LaTeX?
Use \nexists (from amssymb, built in on KaTeX), which renders as the existential quantifier with a slash through it. \neg \exists is an equivalent, slightly longer alternative.
What is the difference between \exists and \forall?
\exists asserts that at least one object satisfies a property; \forall asserts that every object in the domain does. They are logical duals: negating one and pushing the negation inward turns it into the other.