Math 2.0

definition

Uniqueness quantifier

Definition

Let XX be a set and let P(x)P(x) be a predicate on XX. The proposition ∃!x∈X, P(x)\exists! x \in X, \, P(x) is true when exactly one a∈Xa \in X makes P(a)P(a) true. In terms of the other quantifiers, it is the proposition ∃x∈X, (P(x)∧∀y∈X, (P(y)→y=x)).\exists x \in X, \, \bigl( P(x) \land \forall y \in X, \, (P(y) \to y = x) \bigr).

Remarks

A proof of ∃!x, P(x)\exists! x, \, P(x) usually has two parts: existence, an xx with P(x)P(x), and uniqueness, if P(x)P(x) and P(y)P(y) then x=yx = y. For example, ∃!x∈R, 2x+1=7\exists! x \in \mathbb{R}, \, 2x + 1 = 7 is true, with x=3x = 3.

Explanations

L1 human

"There is exactly one even prime" says two things. There is an even prime, namely 22, and no other number is an even prime.

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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What this needs

To read this node, read these 5 nodes in this order. The foundations come first, and the list ends with this node.

Proposition, concept, no evidence levelPropositionconceptPredicate, definition, no evidence levelPredicatedefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinition
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  • no evidence level
  1. Proposition concept
  2. Predicate definition
  3. Existential quantifier definition
  4. Universal quantifier definition
  5. Uniqueness quantifier definition

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348 nodes use this node, directly or through other nodes.

Sources

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

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