more from Herbert B. Enderton

### Single Idea 13201

#### [catalogued under 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 Reference

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]**