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Single Idea 13201

[filed under theme 4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST ]

Full Idea

To know if A ∈ B, we look at the set A as a single object, and check if it is among B's members. But if we want to know whether A ⊆ B then we must open up set A and check whether its various members are among the members of B.

Gist of Idea

∈ says the whole set is in the other; ⊆ says the members of the subset are in the other

Source

Herbert B. Enderton (Elements of Set Theory [1977], 1:04)

Book Ref

Enderton,Herbert B.: 'Elements of Set Theory' [Posts + Telecoms 2006], p.4

A Reaction

This idea is one of the key ideas to grasp if you are going to get the hang of set theory. John ∈ USA ∈ UN, but John is not a member of the UN, because he isn't a country. See Idea 12337 for a special case.

Related Ideas

Idea 12337 There is 'transivity' iff membership ∈ also means inclusion ⊆ [Badiou]

Idea 13443 ∈ relates across layers, while ⊆ relates within layers [Hart,WD]

Idea 15500 Classes divide into subclasses in many ways, but into members in only one way [Lewis]

Idea 15499 A subclass of a subclass is itself a subclass; a member of a member is not in general a member [Lewis]

The 38 ideas from Herbert B. Enderton

13200 Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ [Enderton]
13199 The empty set may look pointless, but many sets can be constructed from it [Enderton]
13201 ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton]
13202 Fraenkel added Replacement, to give a theory of ordinal numbers [Enderton]
13203 The singleton is defined using the pairing axiom (as {x,x}) [Enderton]
13204 The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} [Enderton]
13205 We can only define functions if Choice tells us which items are involved [Enderton]
13206 A 'linear or total ordering' must be transitive and satisfy trichotomy [Enderton]
9718 Validity is either semantic (what preserves truth), or proof-theoretic (following procedures) [Enderton]
9719 A proof theory is 'sound' if its valid inferences entail semantic validity [Enderton]
9720 A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity [Enderton]
9724 Until the 1960s the only semantics was truth-tables [Enderton]
9994 A truth assignment to the components of a wff 'satisfy' it if the wff is then True [Enderton]
9721 A logical truth or tautology is a logical consequence of the empty set [Enderton]
9723 Sentences with 'if' are only conditionals if they can read as A-implies-B [Enderton]
9996 Expressions are 'decidable' if inclusion in them (or not) can be proved [Enderton]
9722 Inference not from content, but from the fact that it was said, is 'conversational implicature' [Enderton]
9995 Proof in finite subsets is sufficient for proof in an infinite set [Enderton]
9997 For a reasonable language, the set of valid wff's can always be enumerated [Enderton]
9712 A relation is 'symmetric' on a set if every ordered pair has the relation in both directions [Enderton]
9713 A relation is 'transitive' if it can be carried over from two ordered pairs to a third [Enderton]
9703 'dom R' indicates the 'domain' of objects having a relation [Enderton]
9705 'fld R' indicates the 'field' of all objects in the relation [Enderton]
9704 'ran R' indicates the 'range' of objects being related to [Enderton]
9710 We write F:A→B to indicate that A maps into B (the output of F on A is in B) [Enderton]
9707 'F(x)' is the unique value which F assumes for a value of x [Enderton]
9715 An 'equivalence relation' is a reflexive, symmetric and transitive binary relation [Enderton]
9716 We 'partition' a set into distinct subsets, according to each relation on its objects [Enderton]
9699 The 'powerset' of a set is all the subsets of a given set [Enderton]
9700 Two sets are 'disjoint' iff their intersection is empty [Enderton]
9702 A 'domain' of a relation is the set of members of ordered pairs in the relation [Enderton]
9701 A 'relation' is a set of ordered pairs [Enderton]
9706 A 'function' is a relation in which each object is related to just one other object [Enderton]
9708 A function 'maps A into B' if the relating things are set A, and the things related to are all in B [Enderton]
9709 A function 'maps A onto B' if the relating things are set A, and the things related to are set B [Enderton]
9711 A relation is 'reflexive' on a set if every member bears the relation to itself [Enderton]
9714 A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects [Enderton]
9717 A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second [Enderton]