Relationships between different types of functions:
A \hookrightarrow B \iff B \twoheadrightarrow A \iff |A| \leq |B|
A \hookrightarrow B \land B \hookrightarrow C \implies A \twoheadrightarrow C
A \leftrightarrow B \iff A \hookrightarrow B \land B \hookrightarrow A
Characterization of countably infinite sets: A is countably infinite iff |A| = |\mathbb{N}| The CS method of showing countability:
A is countable
iff A is encodable
iff each element of A have finite representation
// Exercise: prove A is countably infinite implies |A| = |\mathbb{N}|
Example:
\Sigma^{\infty}: set of all infinite length words (\Sigma^* \cap \Sigma^{\infty} = \emptyset)
// Exercise: Uncountable sets are closed under supersets // Exercise: Practice with uncountability proofs
Predecessor Set: [n] = \{m \in \mathbb{N} | m < n\} Finite: A is finite iff there is a bijection f : [n] \leftrightarrow A
Cardinality: reflexive, symmetric, transitive (equivalence relation)
Countable: can be finite or infinite
Denumerable: countably infinite Recursive Enumerable As Counting Function: a set A \neq 0 is R.E. if there is a function f : \mathbb{N} \rightarrow A such that f is computable.
Theorem: there are only countably many computable functions. (the set of all words is countable, therefore encoded decider turing machine is countable)
Bijection does not imply geometric area: [0, 1] \subseteq \mathbb{R} has te same size as [0, 1]^2 \subseteq R^2
Cardinal: \aleph_0 is the cardinal of \mathbb{N}, \mathbb{Z}, \mathbb{Q}
|\mathbb{R}| = 2^{\aleph_0}
All Cardinals: \aleph_0, \aleph_1, ..., \aleph_n, ... \aleph_w, \aleph_{w^w}, \aleph_{\epsilon_0}, \aleph_{\aleph_1}
Important Ones: \aleph_0, \aleph_1, 2^{\aleph_0}
Cardinal Arithmetic: Let \kappa = |B|, \lambda = |A|
There are uncountable (\mathcal{P}(\mathbb{N})) many binary sequence \mathbb{N} \rightarrow \{0, 1\} (same as characteristic functions subset of natural numbers)
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