\contentsline {part}{I\hspace {1em}Naive Set Theory}{1}{part.1}%
\contentsline {chapter}{\numberline {1}Sets}{3}{chapter.1}%
\contentsline {section}{\numberline {1.1}Sets}{4}{section.1.1}%
\contentsline {section}{\numberline {1.2}Examples of sets}{5}{section.1.2}%
\contentsline {section}{\numberline {1.3}Subsets}{5}{section.1.3}%
\contentsline {section}{\numberline {1.4}Cardinality}{5}{section.1.4}%
\contentsline {section}{\numberline {1.5}Set operations}{6}{section.1.5}%
\contentsline {subsection}{\numberline {1.5.1}Intersection}{6}{subsection.1.5.1}%
\contentsline {subsection}{\numberline {1.5.2}Union}{6}{subsection.1.5.2}%
\contentsline {subsection}{\numberline {1.5.3}Complement}{6}{subsection.1.5.3}%
\contentsline {subsection}{\numberline {1.5.4}Set difference}{6}{subsection.1.5.4}%
\contentsline {subsection}{\numberline {1.5.5}Cartesian product}{7}{subsection.1.5.5}%
\contentsline {subsection}{\numberline {1.5.6}De morgan's laws (set theory)}{7}{subsection.1.5.6}%
\contentsline {section}{\numberline {1.6}Closure operators}{7}{section.1.6}%
\contentsline {section}{\numberline {1.7}Covers}{7}{section.1.7}%
\contentsline {section}{\numberline {1.8}Partitions}{7}{section.1.8}%
\contentsline {subsection}{\numberline {1.8.1}Russel's paradox}{8}{subsection.1.8.1}%
\contentsline {chapter}{\numberline {2}Other collection objects}{9}{chapter.2}%
\contentsline {section}{\numberline {2.1}Multiset}{9}{section.2.1}%
\contentsline {chapter}{\numberline {3}Relation}{11}{chapter.3}%
\contentsline {section}{\numberline {3.1}Relation}{11}{section.3.1}%
\contentsline {subsection}{\numberline {3.1.1}Examples of relations}{11}{subsection.3.1.1}%
\contentsline {section}{\numberline {3.2}Function}{12}{section.3.2}%
\contentsline {section}{\numberline {3.3}Equivalence relation}{13}{section.3.3}%
\contentsline {chapter}{\numberline {4}Functions and Maps}{15}{chapter.4}%
\contentsline {section}{\numberline {4.1}Functions}{15}{section.4.1}%
\contentsline {section}{\numberline {4.2}Types of functions}{16}{section.4.2}%
\contentsline {section}{\numberline {4.3}Indexed families}{17}{section.4.3}%
\contentsline {section}{\numberline {4.4}Tuples and sequences}{18}{section.4.4}%
\contentsline {chapter}{\numberline {5}Cardinality}{19}{chapter.5}%
\contentsline {section}{\numberline {5.1}Countable sets}{19}{section.5.1}%
\contentsline {subsection}{\numberline {5.1.1}Propositions on countable sets}{19}{subsection.5.1.1}%
\contentsline {subsection}{\numberline {5.1.2}Examples of countable sets}{19}{subsection.5.1.2}%
\contentsline {section}{\numberline {5.2}A notable uncountable set}{19}{section.5.2}%
\contentsline {section}{\numberline {5.3}Cardinal numbers}{20}{section.5.3}%
\contentsline {subsection}{\numberline {5.3.1}Aleph numbers}{20}{subsection.5.3.1}%
\contentsline {subsection}{\numberline {5.3.2}Cardinality of the continuum}{21}{subsection.5.3.2}%
\contentsline {subsection}{\numberline {5.3.3}Beth numbers}{21}{subsection.5.3.3}%
\contentsline {subsection}{\numberline {5.3.4}Arithmetic on cardinal numbers}{22}{subsection.5.3.4}%
\contentsline {section}{\numberline {5.4}Ordinal numbers}{23}{section.5.4}%
\contentsline {section}{\numberline {5.5}Large countable numbers}{23}{section.5.5}%
\contentsline {part}{II\hspace {1em}Formalized Set Theory}{25}{part.2}%
\contentsline {chapter}{\numberline {6}Zermelo-Fraenkel Set Theory (ZFC)}{27}{chapter.6}%
\contentsline {section}{\numberline {6.1}Axiom of choice}{27}{section.6.1}%
\contentsline {subsection}{\numberline {6.1.1}Zermelo's theorem}{27}{subsection.6.1.1}%
\contentsline {subsection}{\numberline {6.1.2}Zorn's lemma}{27}{subsection.6.1.2}%