9.7: Summary

• Important definitions:
  • cardinality
  • finite, infinite
  • countable, countably infinite
  • uncountable
• $A$ and $B$ have the same cardinality iff there is a bijection from $A$ to $B$.
• Pigeonhole Principle
• For finite sets $A$ and $B$, we have \(\#(A \times B)=\# A \cdot \# B).
• Inclusion-Exclusion: \(A \cup B=\# A+\# B-\#(A \cap B)).
• Properties of countable sets, including:
  • a countable union of countable sets is countable; and
  • the cartesian product of two countable sets is countable.
• $\mathbb{N}$, $\mathbb{Z}$, and \(\mathbb{Q}) are countable, but \(\mathbb{R}) is uncountable.
• The power set $\mathcal{P}(A)$ has larger cardinality than $A$, for any set $A$.
• Notation:
  • \(\# A)
  • intervals $(a, b)$, $[a, b]$, $[a, b)$, $(a, b]$
  • power set $\mathcal{P}(A)$