9.7: Summary
- Page ID
- 62329
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- 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 \(\mathcal{P}(A)\)