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Single Idea 15500
[filed under theme 4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
]
Full Idea
A class divides exhaustively into subclasses in many different ways; whereas a class divides exhaustively into members in only one way.
Gist of Idea
Classes divide into subclasses in many ways, but into members in only one way
Source
David Lewis (Parts of Classes [1991], 1.2)
Book Ref
Lewis,David: 'Parts of Classes' [Blackwell 1991], p.5
Related Ideas
Idea 13443
∈ relates across layers, while ⊆ relates within layers [Hart,WD]
Idea 13201
∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton]
Idea 12337
There is 'transivity' iff membership ∈ also means inclusion ⊆ [Badiou]
Idea 15499
A subclass of a subclass is itself a subclass; a member of a member is not in general a member [Lewis]
The
40 ideas
with the same theme
[main concepts which are distinctive of set theory]:
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13201
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∈ says the whole set is in the other; ⊆ says the members of the subset are in the other
[Enderton]
|
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13204
|
The 'ordered pair' <x,y> is defined to be {{x}, {x,y}}
[Enderton]
|
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13206
|
A 'linear or total ordering' must be transitive and satisfy trichotomy
[Enderton]
|
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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]
|
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9702
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A 'domain' of a relation is the set of members of ordered pairs in the relation
[Enderton]
|
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9701
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A 'relation' is a set of ordered pairs
[Enderton]
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9706
|
A 'function' is a relation in which each object is related to just one other object
[Enderton]
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9708
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A function 'maps A into B' if the relating things are set A, and the things related to are all in B
[Enderton]
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9709
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A function 'maps A onto B' if the relating things are set A, and the things related to are set B
[Enderton]
|
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9711
|
A relation is 'reflexive' on a set if every member bears the relation to itself
[Enderton]
|
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9714
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A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects
[Enderton]
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9717
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A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second
[Enderton]
|
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9712
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A relation is 'symmetric' on a set if every ordered pair has the relation in both directions
[Enderton]
|
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9713
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A relation is 'transitive' if it can be carried over from two ordered pairs to a third
[Enderton]
|
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12337
|
There is 'transivity' iff membership ∈ also means inclusion ⊆
[Badiou]
|
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15500
|
Classes divide into subclasses in many ways, but into members in only one way
[Lewis]
|
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15499
|
A subclass of a subclass is itself a subclass; a member of a member is not in general a member
[Lewis]
|
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18194
|
'Forcing' can produce new models of ZFC from old models
[Maddy]
|
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9695
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An 'ordered pair' (or ordered n-tuple) is a set with its members in a particular order
[Priest,G]
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9696
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A 'cartesian product' of sets is the set of all the n-tuples with one member in each of the sets
[Priest,G]
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9686
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A 'set' is a collection of objects
[Priest,G]
|
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9687
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A 'member' of a set is one of the objects in the set
[Priest,G]
|
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9688
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A 'singleton' is a set with only one member
[Priest,G]
|
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9689
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The 'empty set' or 'null set' has no members
[Priest,G]
|
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9690
|
A set is a 'subset' of another set if all of its members are in that set
[Priest,G]
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9691
|
A 'proper subset' is smaller than the containing set
[Priest,G]
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9694
|
The 'relative complement' is things in the second set not in the first
[Priest,G]
|
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9693
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The 'intersection' of two sets is a set of the things that are in both sets
[Priest,G]
|
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9692
|
The 'union' of two sets is a set containing all the things in either of the sets
[Priest,G]
|
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9698
|
The 'induction clause' says complex formulas retain the properties of their basic formulas
[Priest,G]
|
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10890
|
A 'partial ordering' is reflexive, antisymmetric and transitive
[Zalabardo]
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10889
|
The 'Cartesian Product' of two sets relates them by pairing every element with every element
[Zalabardo]
|
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10098
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The 'power set' of A is all the subsets of A
[George/Velleman]
|
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10099
|
The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}}
[George/Velleman]
|
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10101
|
Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B
[George/Velleman]
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10859
|
A set is 'well-ordered' if every subset has a first element
[Clegg]
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15914
|
An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one
[Lavine]
|
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8665
|
A 'proper subset' of A contains only members of A, but not all of them
[Friend]
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8672
|
A 'powerset' is all the subsets of a set
[Friend]
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