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}
Countable: A set S is called countable iff S is either finite or countably infinite.
Uncountably infinite: |A| > |\mathbb{N}|
\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.
\text{There exists an injection} f : A \hookrightarrow B \iff \text{There exists an surjection} g : B \rightarrowtail A.
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