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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]
| 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] |