Organization:Foundational relation
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In set theory, a foundational relation on a set or proper class lets each nonempty subset admit a relational minimal element.
Formally, let (A, R) be a binary relation structure, where A is a class (set or proper class), and R is a binary relation defined on A. Then (A, R) is a foundational relation if any nonempty subset in A has an R-minimal element. In predicate logic,
- [math]\displaystyle{ (\forall S)\left(S \subseteq A \land S \not= \emptyset \Rightarrow (\exists x \in S)(\forall y \in S)(\lnot y R x)\right), }[/math] [1]
in which [math]\displaystyle{ \emptyset }[/math] denotes the empty set. Here [math]\displaystyle{ x }[/math] is an R-minimal element in the subset S, since none of its R-predecessors is in S.
See also
References
- ↑ See Definition 6.21 in Zaring W.M., G. Takeuti (1971). Introduction to axiomatic set theory (2nd, rev. ed.). New York: Springer-Verlag. ISBN 0387900241.
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