Elements of Set Theory¶
Enderton, H. B. (1977). Elements of Set Theory. Academic Press.
Cited by¶
2 citations across 2 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Equivalence Relation
- 1.
This sourceDevelops equivalence relations, equivalence classes [a]_R, the quotient set A/R, and the theorem that A/R partitions A.
- 1.
- Well-Foundedness (Well-Ordering)
- Well-foundedness delivers three mutually-equivalent reasoning patterns, as Enderton (1977) develops in his standard introduction to set theory: (1) well-founded induction (to prove ∀x: P(x) on a well-founded domain, prove ∀x: (∀y ≺ x: P(y)) ⇒ P(x) — no explicit base case needed because minimal elements have no predecessors, so the inductive step degenerates into the base case for them); (2) well-founded recursion (to define f(x) on a well-founded domain, define it in terms of f on ≺-predecessors — the definition terminates because there is no infinite descent, and the function is uniquely determined by its recursive equation); (3) minimal-counterexample method (to prove ∀x: P(x), assume a counterexample exists, take a ≺-minimal counterexample, derive a contradiction) — all three patterns rely on the same no-infinite-descent structure.
This sourceFoundational set-theory textbook: develops the carrier-relation-axioms-class-quotient-use breakdown of equivalence-relation structure with explicit attention to partition-quotient correspondence.
- Well-foundedness delivers three mutually-equivalent reasoning patterns, as Enderton (1977) develops in his standard introduction to set theory: (1) well-founded induction (to prove ∀x: P(x) on a well-founded domain, prove ∀x: (∀y ≺ x: P(y)) ⇒ P(x) — no explicit base case needed because minimal elements have no predecessors, so the inductive step degenerates into the base case for them); (2) well-founded recursion (to define f(x) on a well-founded domain, define it in terms of f on ≺-predecessors — the definition terminates because there is no infinite descent, and the function is uniquely determined by its recursive equation); (3) minimal-counterexample method (to prove ∀x: P(x), assume a counterexample exists, take a ≺-minimal counterexample, derive a contradiction) — all three patterns rely on the same no-infinite-descent structure.
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