Lecture 006

Ada's Lecture

Countable and Uncountable Sets

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:

• The set of all polynomials in one variable with rational coefficients is countable.

$\Sigma^{\infty}$: set of all infinite length words ($\Sigma^* \cap \Sigma^{\infty} = \emptyset$)

• there is a surjection from $A$ to $\{0, 1\}^{\infty}$, then $A$ is uncountable. (or subset of $A$ has a bijection with $\{0, 1\}^{\infty}$)

// Exercise: Uncountable sets are closed under supersets // Exercise: Practice with uncountability proofs

Sutner's Lecture

Cardinality

Predecessor Set: $[n] = \{m \in \mathbb{N} | m < n\}$ Finite: $A$ is finite iff there is a bijection $f : [n] \leftrightarrow A$

• for finite $\text{injective} \equiv \text{surjective} \equiv \text{bijective}$

Cardinality: reflexive, symmetric, transitive (equivalence relation)

• // WARNING: not a preorder since $|A|\leq|B| \land |B| \leq |A| \;\not\!\!\!\implies A = B$

Countability

Countable: can be finite or infinite

• closed under finite and countably finite unions

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 $\kappa = |B|, \lambda = |A|$

  • $\kappa = \kappa + \kappa = \kappa \cdot \kappa = \kappa$
  • $\kappa + \lambda = \kappa \cdot \lambda = \max(\kappa, \lambda)$
  • $\kappa^\lambda = |A \rightarrow B|$
  • Continumm Hypothesis: $2^{\aleph_0} = \aleph_1$ (but this result could go either way since one can choose $2^{\aleph_0}$)

• There are uncountable ($\mathcal{P}(\mathbb{N})$) many binary sequence $\mathbb{N} \rightarrow \{0, 1\}$ (same as characteristic functions subset of natural numbers)