Math 2.0

definition

Universal quantifier

Definition

Let XX be a set and let P(x)P(x) be a predicate on XX. The universal quantification of PP over XX is the proposition ∀x∈X, P(x)\forall x \in X, \, P(x), read "for all xx in XX, P(x)P(x)". It is true when P(a)P(a) is true for each a∈Xa \in X, and false otherwise.

Remarks

When XX is empty, ∀x∈X, P(x)\forall x \in X, \, P(x) is true for each PP. This is a case of vacuous truth. 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) \land \dots \land P(a_n).

Explanations

L1 human

"Every square of a real number is at least 00" is ∀x∈R, x2≥0\forall x \in \mathbb{R}, \, x^2 \ge 0. To prove it, take an arbitrary xx and show x2≥0x^2 \ge 0. To refute it, one counterexample is enough.

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 levelPredicatedefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinition
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  1. Proposition concept
  2. Predicate definition
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Sources

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

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