Set Theory

Jan Hubička, hubicka@kam.mff.cuni.cz

Consultations by email arrangement. Use "Set Theory" in subjects of emails.

Topics covered

Feb 16
Feb 23
Mar 2
Mar 9
Mar 16
Cardinality of sets, Cantor-Bernstein
  • Def: Sets A and B are equipotent (have the same cardinality) if there exists one-to-one functions with domain A and range B. In this case we write |A|=|B|.
  • Theorem:
    • \(|A|=|A|\)
    • \(|A|=|B|\implies |B|=|A|\)
    • \((|A|=|B|)=|C| \implies |A|=|C|\)
  • Being equipotent is an equivalence "relation". Not in the sense we defined relations before because its domain is the class of all sets, but we can speak of class relations.
  • Def: \(|A|\leq |B|\) if there exists \(f:A\to B\) that is injective.
  • Theorem:
    • \(|A|\leq |A|\)
    • \(|A|=|B|\leq |C|=|D|\implies |A|\leq |C|\) and \(|B|\leq |D||\)
    • \(|A|\leq |B|\leq |C|\implies |A|\leq |C|\)
  • We know that $\leq$ is an order. Is it linear? This is an important theorem.
  • Theorem (Cantor-Bernstein) \(|X|\leq |Y|\) and \(|Y|\leq |X|\) implies \(|X|=|Y|\).
  • Proof using the following Lemma: If \(A_1\subseteq B\subseteq A\) and \(|A_1|=|A|\) then \(|A|=|B|\). Proof of lemma is quite smart and requires careful definition of the bijection.
  • Mar 23
    Finite sets
    Countable sets
    Mar 30
    Cardinal numbers
    Apr 6
    Well ordered sets
    Apr 13
    Ordinal numbers
    Apr 20
    Apr 27
    May 6
    May 17

    Literature

    Many recommended books are mentioned in the SIS description of the class. We loosely follow the book by Hrbacek and Jech. You may want to consult notes of Václav Končický (in Czech).