Sets
Almost every object in mathematics is a collection of other objects. What exactly is a collection, and what can be built from collections?
Where you are 15 inside
What this area is about
A set is decided entirely by its members: an object is in it or not. From that one idea come operations that build new sets, ways to compare and count sets, and the functions and relations that connect one set to another. Much of later mathematics is written in this language.
How these fit together
Definitions. Elements and membership, roster and rule notation, subsets and equal sets, cardinality |A| of a finite set, and the power set \mathcal{P}(A).
Building sets. Union, intersection and complement and the Cartesian product A \times B.
Results and proofs. The distributive and De Morgan laws, proving a set identity by showing each side is a subset of the other, and counting a union: for finite sets, |A \cup B| = |A| + |B| - |A \cap B|.
Relations and functions. Properties of relations lead to equivalence classes. One-to-one and onto functions lead to composition and inverses. Infinite sets compare sizes by bijections.
A path through it
Elements first, then roster notation and subsets. Roster notation opens cardinality, power sets, products and union and intersection. Union leads to the set laws (with De Morgan’s laws from logic) and to counting a union. Proving a set identity needs subsets and direct proof. Products lead to relations; equivalence classes come after relations, counting a union and congruence from number theory. Subsets lead to one-to-one and onto functions, then composition. Infinite sets come after cardinality, power sets and bijections.