Naive Set Theory Axioms — Studies in Algebra and Group Theory

N1 Naive Set Theory Axioms

The axioms

Definition N1.1 (I. Axiom of extension). Two sets are equal iff they have the same elements.

Definition N1.2 (II. Axiom of specification). For every set and every assertion , there exists a set which is all the such that is true. is called a subset of .

Definition N1.3 (III. Axiom of pairs). For any two sets there exists a set that they both belong to.

Definition N1.4 (IV. Axiom of unions). For every collection of sets there exists a set that contains all the elements of at least one set of the given collection.

Note (Union notation).

Consequences and terms

Set pair generation comes from .
: and are disjoint.
If any two distinct sets in a collection are disjoint, they are pairwise disjoint (a pairwise disjoint collection).
Distributive law: , and the inverse (with and swapped): .
Difference relative complement.
Under difference, is our universe, such that , .
De Morgan laws.
Principle of duality for sets.
Relation: a set of ordered pairs (cf. functions).
- reflexive (duh): ;
- symmetric: ;
- transitive (duh).