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Single Idea 9702
[filed under theme 4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
]
Full Idea
The 'domain' of a relation is the set of all objects that are members of ordered pairs that are members of the relation.
Gist of Idea
A 'domain' of a relation is the set of members of ordered pairs in the relation
Source
Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0)
Book Ref
Enderton,Herbert B.: 'A Mathematical Introduction to Logic' [Academic Press 2001], p.4
The
40 ideas
with the same theme
[main concepts which are distinctive of set theory]:
13201
|
∈ says the whole set is in the other; ⊆ says the members of the subset are in the other
[Enderton]
|
13204
|
The 'ordered pair' <x,y> is defined to be {{x}, {x,y}}
[Enderton]
|
13206
|
A 'linear or total ordering' must be transitive and satisfy trichotomy
[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]
|
9713
|
A relation is 'transitive' if it can be carried over from two ordered pairs to a third
[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]
|
9712
|
A relation is 'symmetric' on a set if every ordered pair has the relation in both directions
[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]
|
9711
|
A relation is 'reflexive' on a set if every member bears the relation to itself
[Enderton]
|
12337
|
There is 'transivity' iff membership ∈ also means inclusion ⊆
[Badiou]
|
15500
|
Classes divide into subclasses in many ways, but into members in only one way
[Lewis]
|
15499
|
A subclass of a subclass is itself a subclass; a member of a member is not in general a member
[Lewis]
|
18194
|
'Forcing' can produce new models of ZFC from old models
[Maddy]
|
9695
|
An 'ordered pair' (or ordered n-tuple) is a set with its members in a particular order
[Priest,G]
|
9696
|
A 'cartesian product' of sets is the set of all the n-tuples with one member in each of the sets
[Priest,G]
|
9687
|
A 'member' of a set is one of the objects in the set
[Priest,G]
|
9686
|
A 'set' is a collection of objects
[Priest,G]
|
9688
|
A 'singleton' is a set with only one member
[Priest,G]
|
9689
|
The 'empty set' or 'null set' has no members
[Priest,G]
|
9691
|
A 'proper subset' is smaller than the containing set
[Priest,G]
|
9690
|
A set is a 'subset' of another set if all of its members are in that set
[Priest,G]
|
9692
|
The 'union' of two sets is a set containing all the things in either of the sets
[Priest,G]
|
9693
|
The 'intersection' of two sets is a set of the things that are in both sets
[Priest,G]
|
9694
|
The 'relative complement' is things in the second set not in the first
[Priest,G]
|
9698
|
The 'induction clause' says complex formulas retain the properties of their basic formulas
[Priest,G]
|
10889
|
The 'Cartesian Product' of two sets relates them by pairing every element with every element
[Zalabardo]
|
10890
|
A 'partial ordering' is reflexive, antisymmetric and transitive
[Zalabardo]
|
10098
|
The 'power set' of A is all the subsets of A
[George/Velleman]
|
10099
|
The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}}
[George/Velleman]
|
10859
|
A set is 'well-ordered' if every subset has a first element
[Clegg]
|
15914
|
An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one
[Lavine]
|
8665
|
A 'proper subset' of A contains only members of A, but not all of them
[Friend]
|
8672
|
A 'powerset' is all the subsets of a set
[Friend]
|