Math 2.0

definition

Existential quantifier

Definition

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

Remarks

When XX is empty, ∃x∈X, P(x)\exists x \in X, \, P(x) is false for each PP. For a finite set X={a1,…,an}X = \{a_1, \dots, a_n\}, the statement is equivalent to P(a1)∨⋯∨P(an)P(a_1) \lor \dots \lor P(a_n). The uniqueness quantifier ∃!\exists! says that exactly one element exists.

Explanations

L1 human

"There is an even prime number" is ∃n∈N, (n is even∧n is prime)\exists n \in \mathbb{N}, \, (n \text{ is even} \land n \text{ is prime}). To prove it, one example is enough: n=2n = 2.

What this node needs for W2
  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

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Proposition, concept, no evidence levelPropositionconceptPredicate, definition, no evidence levelPredicatedefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinition
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  1. Proposition concept
  2. Predicate definition
  3. Existential quantifier definition

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Sources

  1. Web https://en.wikipedia.org/wiki/Existential_quantification, lead section

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