Existential quantifier

Definition

An existential quantifier is a symbol used in mathematical logic to express that there exists at least one element in a domain which satisfies a given property. It is denoted by the symbol $\exists$.

5 Must Know Facts For Your Next Test

1. The existential quantifier $\exists$ is typically used in conjunction with variables and predicates to form logical statements.
2. In the precise definition of a limit, the existence of certain values fulfilling specific conditions is often expressed using $\exists$.
3. A statement involving an existential quantifier has the general form $\exists x \, P(x)$, which reads as 'there exists an x such that P(x)'.
4. Existential quantifiers are foundational in defining limits rigorously by indicating the presence of values within specified bounds.
5. When proving limits, showing that some $\delta$ or $\epsilon$ exists to satisfy conditions often involves using existential quantifiers.

Review Questions

• What does the symbol $\exists$ represent in mathematical logic?
• How is an existential quantifier used in the precise definition of a limit?
• Can you provide an example of a statement involving an existential quantifier?

Related terms

Universal Quantifier: A symbol ($\forall$) used to denote that all elements in a domain satisfy a given property.

Predicate: A function or relation that returns true or false for given inputs.

$\epsilon-\delta$ Definition of Limit: A formal definition of a limit using two small positive numbers, $\epsilon$ (epsilon) and $\delta$ (delta), to rigorously define what it means for a function to approach a limit.