Mathematical logic and foundations Definition

Existential quantifier

Definition

Let $X$ be a set and let $P(x)$ be a predicate on $X$ . The existential quantification of $P$ over $X$ is the proposition $\exists x \in X, \, P(x)$ , read "there exists $x$ in $X$ such that $P(x)$ ". It is true when $P(a)$ is true for at least one $a \in X$ , and false otherwise.

Remarks

When $X$ is empty, $\exists x \in X, \, P(x)$ is false for each $P$ . For a finite set $X = \{a_1, \dots, a_n\}$ , the statement is equivalent to $P(a_1) \lor \dots \lor P(a_n)$ . The uniqueness quantifier $\exists!$ says that exactly one element exists.

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Proposition concept
Predicate definition
Existential quantifier definition

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Tag	`0JW2FHW7G3`
URI	`m2://0JW2FHW7G3`
Kind	definition
Writing	W1 stated
MSC	03B10
Defines	Existential quantifier
Uses	Predicate

Existential quantifier

Definition

Remarks

Reading path

Where this leads

Further reading

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