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### 4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST

#### [main concepts which are distinctive of set theory]

40 ideas
 13201 ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton]
 13204 The 'ordered pair' 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]
 9708 A function 'maps A into B' if the relating things are set A, and the things related to are all in B [Enderton]
 9701 A 'relation' is a set of ordered pairs [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]
 9706 A 'function' is a relation in which each object is related to just one other object [Enderton]
 9711 A relation is 'reflexive' on a set if every member bears the relation to itself [Enderton]
 9709 A function 'maps A onto B' if the relating things are set A, and the things related to are set B [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]
 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]
 9696 A 'cartesian product' of sets is the set of all the n-tuples with one member in each of the sets [Priest,G]
 9695 An 'ordered pair' (or ordered n-tuple) is a set with its members in a particular order [Priest,G]
 9686 A 'set' is a collection of objects [Priest,G]
 9687 A 'member' of a set is one of the objects in the set [Priest,G]
 9689 The 'empty set' or 'null set' has no members [Priest,G]
 9688 A 'singleton' is a set with only one member [Priest,G]
 9690 A set is a 'subset' of another set if all of its members are in that set [Priest,G]
 9691 A 'proper subset' is smaller than the containing set [Priest,G]
 9694 The 'relative complement' is things in the second set not in the first [Priest,G]
 9693 The 'intersection' of two sets is a set of the things that are in both sets [Priest,G]
 9692 The 'union' of two sets is a set containing all the things in either of the sets [Priest,G]
 9698 The 'induction clause' says complex formulas retain the properties of their basic formulas [Priest,G]
 10890 A 'partial ordering' is reflexive, antisymmetric and transitive [Zalabardo]
 10889 The 'Cartesian Product' of two sets relates them by pairing every element with every element [Zalabardo]
 10098 The 'power set' of A is all the subsets of A [George/Velleman]
 10099 The 'ordered pair' , for two sets a and b, is the set {{a, b},{a}} [George/Velleman]
 10101 Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B [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]