Naive Set Theory¶
Halmos, P. R. (1960). Naive Set Theory. Springer.
Cited by¶
22 citations across 22 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Asymmetry
- Many inequalities are asymmetries, but many asymmetries are not inequalities, a distinction Halmos (1974) draws cleanly in the relational set-theory treatment that separates ordering structure from quantitative comparison.
This sourceDevelops ordered pairs, relations, and partial orders in the Kuratowski formalization, defining antisymmetry (xRy and yRx imply x=y) as a qualitative ordering property separable from any quantitative comparison on a shared metric.
- Many inequalities are asymmetries, but many asymmetries are not inequalities, a distinction Halmos (1974) draws cleanly in the relational set-theory treatment that separates ordering structure from quantitative comparison.
- Bijectivity
- When both hold, the correspondence is reversible: there is a well-defined inverse map that recovers the source from the target without loss or ambiguity.
This sourceStandard reference defining injective, surjective, and bijective (one-to-one and onto) functions, the existence and uniqueness of the inverse of a bijection, and equinumerosity / cardinality via bijection.
- When both hold, the correspondence is reversible: there is a well-defined inverse map that recovers the source from the target without loss or ambiguity.
- Complement
- Formally, given a universe \(U\) and a subset \(A \subseteq U\), the complement \(A^c\) (equivalently \(U \setminus A\)) is the set \(\{x \in U : x \notin A\}\).
This sourceStandard introduction defining set complement, relative complement, and the Boolean algebra of subsets of a universe. (
- Formally, given a universe \(U\) and a subset \(A \subseteq U\), the complement \(A^c\) (equivalently \(U \setminus A\)) is the set \(\{x \in U : x \notin A\}\).
- Disjoint union
- Disjointness
- Two collections are disjoint when they share no element: their intersection is empty.
This sourceStandard reference defining intersection and disjointness: two sets are disjoint exactly when their intersection is the empty set, the strong no-shared-element relation under a fixed membership criterion.
- Two collections are disjoint when they share no element: their intersection is empty.
- Empty Set
- In mathematics it is ∅ in set theory, the zero ideal, the empty product (one) and empty sum (zero), and the vacuous base of induction.
This sourceDefines the empty set ∅ as a fully constructed object — the identity of union (A ∪ ∅ = A), the base of inductive constructions, and the seat of vacuous truth (every member of ∅ has any property), with the empty product equal to one.
- In mathematics it is ∅ in set theory, the zero ideal, the empty product (one) and empty sum (zero), and the vacuous base of induction.
- Equivalence Relation
- … — formally, a binary relation $\sim$ on a set $S$ that is reflexive ($a \sim a$ for every $a \in S$), symmetric ($a \sim b$ implies $b \sim a$ for every $a, b \in S$), and transitive ($a \sim b$ and $b \sim c$ together imply $a \sim c$ for every $a, b, c \in S$), in the canonical formulation given by Halmos (1960).
This sourceGives the canonical axiomatic definition of an equivalence relation (reflexive, symmetric, transitive) and the partition-equivalence theorem (every equivalence relation determines a partition and conversely).
- … — formally, a binary relation $\sim$ on a set $S$ that is reflexive ($a \sim a$ for every $a \in S$), symmetric ($a \sim b$ implies $b \sim a$ for every $a, b \in S$), and transitive ($a \sim b$ and $b \sim c$ together imply $a \sim c$ for every $a, b, c \in S$), in the canonical formulation given by Halmos (1960).
- Function (Mapping)
- This is the core structural commitment that distinguishes functions from general relations
This sourceVan Nostrand. Standard introductory set-theory text: gives the canonical axiomatic definition of an equivalence relation as a reflexive, symmetric, and transitive binary relation, together with the partition-equivalence theorem.
- This is the core structural commitment that distinguishes functions from general relations
- Indexed family
- Intersection
- In mathematics and logic it appears as set intersection, the conjunction of predicates, the meet operation in lattice theory, the kernel of a system of constraints, and the feasible region of an integer program defined by AND-of-conditions.
This sourceStandard reference defining set intersection, its associative/commutative/idempotent algebra, and De Morgan duality with union.
- In mathematics and logic it appears as set intersection, the conjunction of predicates, the meet operation in lattice theory, the kernel of a system of constraints, and the feasible region of an integer program defined by AND-of-conditions.
- Partition
- In mathematics it is the formal partition of a set, interchangeable with an equivalence relation (each induces the other), and the partition of a probability space into events for the law of total probability.
This sourceStandard reference for partitions of a set and their one-to-one correspondence with equivalence relations.
- In mathematics it is the formal partition of a set, interchangeable with an equivalence relation (each induces the other), and the partition of a probability space into events for the law of total probability.
- Predicate
- Write \(S = \{\, n \in \mathbb{N} : n \text{ is prime} \,\}\).
This sourceTreats set-builder notation defining a set as the extension of a predicate over a domain.
- Write \(S = \{\, n \in \mathbb{N} : n \text{ is prime} \,\}\).
- Surjectivity
- A map can be injective without being surjective (a small set embedded into a large one, leaving most targets unhit) and surjective without being injective (a large set folded onto a small one, hitting every target many times over).
This sourceDefines onto (surjective) functions, the image as a subset of the codomain, right inverses, and the existence of a section for a surjection.
- A map can be injective without being surjective (a small set embedded into a large one, leaving most targets unhit) and surjective without being injective (a large set folded onto a small one, hitting every target many times over).
- Union
- In set theory and combinatorics the inclusion–exclusion principle is built on union: the size of a union of overlapping sets is the alternating sum that corrects for multiply-counted elements, the canonical formula for "how big is the combined set?"
This sourceStandard reference defining the union of sets by the at-least-one-membership (inclusive-OR) test and its associative, commutative, idempotent, and monotone properties.
- In set theory and combinatorics the inclusion–exclusion principle is built on union: the size of a union of overlapping sets is the alternating sum that corrects for multiply-counted elements, the canonical formula for "how big is the combined set?"
- Well-Foundedness (Well-Ordering)
- 1.
This sourceVan Nostrand. Standard introductory set-theory text: gives the canonical axiomatic definition of an equivalence relation as a reflexive, symmetric, and transitive binary relation, together with the partition-equivalence theorem.
- 1.
Domain-specific¶
Mechanisms¶
- Deduplicating Union Pass
- Its strength is that it makes the union idempotent and honest: run it twice, feed it the same source twice, and the member count does not move, because union is idempotent
This sourceStates the idempotence law A ∪ A = A, so repeated appearances of the same set do not change the union.
- Its strength is that it makes the union idempotent and honest: run it twice, feed it the same source twice, and the member count does not move, because union is idempotent
- Reverse Mapping Table
- Its failure mode is arbitrary preimage selection creeping in at the consumer: a reader glances at an ambiguous row, takes the first entry, and treats a set-valued inverse as single-valued.
This sourceDefines an inverse image as the set of all inputs mapped into a target and limits a single-valued inverse to one-to-one functions.
- Its failure mode is arbitrary preimage selection creeping in at the consumer: a reader glances at an ambiguous row, takes the first entry, and treats a set-valued inverse as single-valued.
Verification¶
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Links previously used in the corpus¶
Before the registry existed this work was also linked 4 other ways.
- https://link.springer.com/book/10.1007/978-1-4757-1645-0 ×4
- https://archive.org/details/naivesettheory0000paul ×2
- https://search.worldcat.org/title/Naive-set-theory/oclc/477189496 ×2
- https://archive.org/details/naivesettheory0000halm ×1
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