Lecture 006

Ada's Lecture

Countable and Uncountable Sets

Injection, Surjection, Bijection

Injection, Surjection, Bijection



Power Sets

Power Sets

Number Sets

Number Sets

Relationships between different types of functions:

Characterization of countably infinite sets: A is countably infinite iff |A| = |\mathbb{N}| The CS method of showing countability:

// Exercise: prove A is countably infinite implies |A| = |\mathbb{N}|


\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

Sutner's Lecture


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}

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