Lecture 020

Cardinality on Sets

|A| \leq |B| \iff (\exists \text{injection} A \hookrightarrow B)
|A| \geq |B| \iff (\exists \text{surjection} A \rightarrowtail B)
|A| = |B| \iff (\exists \text{bijection} A \to B)

Cartesian Product of Countable Sets

  1. If A and B are finite, then |A\times B| = |A| \dot |B| (unassigned homework)
  2. If |A| = \mathcal{N}_0 \land \text{B is finite} \implies |A\times B| = \mathcal{N}_0 (recitation)

  3. e.g. |\mathbb{N} \times [2]| = |\mathbb{N}|

  4. |A| = \mathcal{N}_0 \land |B| = \mathcal{N}_0 \implies |A\times B| = \mathcal{N}_0 (unassigned homework+old recitation)

  5. e.g. |\mathbb{N}^2| = |\mathbb{N}|

  6. If \{A_i\}_{i \in [n]} is a finite collection or countably infinite set, then A_1 \times ... \times A_n = \prod_{i=1}^n {A_i} is a countably infinite (3 + induction)

Theorem: The union of countably infinite set \{A_n\}_{n\in \mathbb{N}} is countably infinite.


  1. If \{A_i\}_{i\in [n]} is a finite family of countably infinite sets, then the union is countably infinite
  2. If \{A_i\}_{i\in \mathbb{N}} is a family of countable sets then the union is still countable

Table of Content