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.

• proof by double containment:
• WTS $|\mathbb{N}| \leq |A|$. We know there exists a bijection from N to A1. Then there is a injection from N to A1. Then there is a injection from N to A.
• WTS $|\mathbb{N}| \geq |A|$. We extract an element from A, find out its belonging set (in A_n). $h: \mathbb{N}^2 \rightarrowtail A | h(n, m_ = a(n, m)$. Then there exists m such that y=a(n, m)=h(n, m). Since $|\mathbb{N}^2| geq |A|$, $|A| \leq |N|$ TODO:???

Corollaries

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

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