Lecture 019

Cardinality

Cardinality: the cardinality of a set S, denoted $|S|$, is the number of elements in the set.

Same Cardinality: S and T are said to have the same cardinality ($|S| = |T|$) iff $(\exists bijection f: S \to T)$

Finite: A set S is finite (denoted $|S| = n$) iff $(\exists n \in \mathbb{N})(f:[n] \to S \text{ is a bijection})$

• IMPORTANT: recall that $[0] = \emptyset$

• $(\forall m, n \in \mathbb{N})(m\neq n \implies \lnot (|S| = m \land |S| = n))$ because $[m] \to [n]$ is not a bijection.

• because if so, then $[n] \ \{ \text{all result of [m]\to[S]\to[n]} \} \neq \emptyset$, contradicting surjectivity

• see lecture note for more detail

Infinite: A set S is called infinite iff S is not finite.

Countably infinite (list-able set, enumerable set): A set S is called countably infinite iff there exists a bijection $f: \mathbb{N} \to S$ denoted $|S| = \mathcal{N}_0$ or $|S| = |\mathbb{N}| = \mathcal{N}_0$

• including $\mathbb{N}, 2\mathbb{N}, \mathbb{Z}, \mathbb{N}^2$

• including $\mathbb{Q}$

  • we show it by snaking proof. (sip repeat if wanted to show bijection, include repeat if wanted to show surjection)
  • we show it by CBS: there is a surjection f(x) = x/1. Define Q as $Q = \{\frac{a}{b}|a\in \mathbb{Z}, b \in \mathbb{Z}^+, gcd(a,b)=1\}$ Define $g(a/b)=\{2^a5^b \text{if a >= 0}, 3^{-a}5^b \text{if a < 0}\}$ then, in 3 cases, show injection by Fundamental Theorem of Algebra.

Countable: A set S is called countable iff S is either finite or countably infinite.

Uncountably infinite: $|A| > |\mathbb{N}|$

• there exists no surjection $f:\mathbb{N} \rightarrowtail A$

Schröder–Bernstein theorem CBS Theorem

$\text{There exists an injection} f : A \hookrightarrow B \land \text{There exists an injection} g : B \hookrightarrow A \implies \text{There exists an bijection} h : A \to B$.

Partition Principle

$\text{There exists an injection} f : A \hookrightarrow B \iff \text{There exists an surjection} g : B \rightarrowtail A$.