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Apuntes Rdgz Documentation

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What's inside Apuntes Rdgz Documentation

  1. Overview of Zermelo-Fraenkel (ZF) Set Theory Axioms

    Zermelo-Fraenkel set theory uses the concepts of 'set' and 'membership' as primitives. To avoid paradoxes like Russell's Paradox (where a set is defined as containing all sets that do not contain themselves), ZF limits set construction to specific axiomatic rules. A property alone cannot define a set; it can only be used to select elements from an existing set.

  2. Introduction to sequences of random variables

    In probability theory, a sequence of random variables {Xn}_{n∈ℕ} is analyzed within a probability space (Ω, 𝒜, P) to determine the behavior of the probabilities associated with Xn as n approaches infinity (the limit of the sequence).

  3. Overview of the Gentzen System for Propositional Logic

    The Gentzen system, also known as the Natural Deduction system for propositional logic, is defined by a specific syntax and a set of inference rules. It contains no axioms.

    Syntax Constraints:

    • It follows standard propositional logic syntax with two exceptions:
      1. The constants $\top$ (truth) and $\bot$ (falsity) are not included in the alphabet.
      2. The biconditional $\leftrightarrow$ is not a binary connective; it is treated as an abbreviation for $(\varphi \to \psi) \wedge (\psi \to \varphi)$.

  4. Apply Gentzen inference rules for basic connectives

    The Gentzen system uses introduction (I) and elimination (E) rules to derive formulas. Let $A, B, C$ be any propositional formulas:

    • Negation Elimination ($E \neg$): Allows deriving $A$ from $\neg\neg A$.
    • Conjunction Introduction ($I \wedge$): Allows deriving $A \wedge B$ from $A$ and $B$.
    • Conjunction Elimination ($E \wedge$): Two rules allowing the derivation of $A$ from $A \wedge B$, or $B$ from $A \wedge B$.
    • Disjunction Introduction ($I \vee$): Two rules allowing the derivation of $A \vee B$ from $A$, or $A \vee B$ from $B$.
    • Implication Elimination ($E \to$): Also known as modus ponens; allows deriving $B$ from $A$ and $A \to B$.

  5. Determine the equivalence class of an integer modulo n

    The equivalence class of an integer $a$ modulo $n$, denoted as $[a]_n$, consists of all integers $x$ such that $x \equiv a \pmod{n}$. A fundamental property is that $[a]_n = [r]_n$, where $r$ is the remainder of the division of $a$ by $n$ (from the division theorem $a = nq + r$ with $0 \leq r < n$).
    Example: Find the class of 353 modulo 6
353 = 6 * 58 + 5
Therefore, [353]_6 = [5]_6 because 353 - 5 = 6 * 58.

  6. Understand the Fundamental Theorem of Calculus

    The Fundamental Theorem of Calculus (Teorema VII.3) establishes that if a function $f$ is continuous on the interval $[a, b]$, there exists a function $F$ such that $F'(x) = f(x)$ for all $x ∈ [a, b]$. This function $F$ can be defined as the integral of $f$ from a fixed point $a$ to a variable $x$.
    F(x) = \int_{a}^{x} f(t) \mathrm{d}t

  7. Understand the Central Limit Theorem (CLT) for i.i.d. variables

    The Central Limit Theorem states that if $X_1, X_2, \dots, X_n$ is a sequence of independent and identically distributed (i.i.d.) random variables with a finite mean $\mu = E(X_i)$ and a finite variance $\sigma^2 = V(X_i) < \infty$, the distribution of the sample mean $\overline{X}_n$ converges to a normal distribution as $n$ increases. Specifically, the standardized variable converges in distribution to the standard normal distribution $Z$:

    $$\frac{\overline{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} Z$$

    This applies to both discrete and continuous variables, whether they are symmetric, asymmetric, unimodal, or multimodal.

  8. Understand formal language basics (Alphabet and Words)

    In formal logic and computer science, an alphabet (denoted as $A$) is a set of symbols. A word over $A$ is a finite sequence of symbols from that alphabet. The set of all possible words over $A$ is denoted as $A^$. A language $L$ is defined as a subset of all possible words ($L \subseteq A^$).

  9. Define a $\sigma$-algebra over a sample space

    A $\sigma$-algebra $\mathscr{A}$ over a sample space $\Omega$ is a collection of subsets of $\Omega$ ($\mathscr{A} \subset \mathcal{P}(\Omega)$) that satisfies three fundamental properties:

    1. It contains the certain event: $\Omega \in \mathscr{A}$.
    2. It is closed under complementation: If $A \in \mathscr{A}$, then $A^c \in \mathscr{A}$.
    3. It is closed under countable unions: If a sequence of sets ${A_n}{n \in \mathbb{N}}$ is in $\mathscr{A}$, then $\bigcup{n=1}^{\infty} A_n \in \mathscr{A}$.

  10. Define and identify discrete random vectors

    A random vector $X = (X_1, \dots, X_n)$ is considered discrete if every component $X_i$ (for $i=1, 2, \dots, n$) is a discrete random variable. Consequently, the support of a discrete random vector—the set of values it can take in $\mathbb{R}^n$—is countable.

  11. Apply Double Negation (DN) in Gentzen system

    The Double Negation (DN) rule allows the introduction of a double negation to a formula.

    Formal Statement: {𝐴} ⊢ ¬¬𝐴

  12. Understand the relationship between equivalence relations and partitions

    There is a fundamental duality between equivalence relations and partitions of a set:

    1. Equivalence Relation $\rightarrow$ Partition: If $R$ is an equivalence relation on a set $A$, the set of all equivalence classes $[a] = {x \in A \mid aRx}$ forms a partition of $A$. This means the classes are pairwise disjoint (any two distinct classes have no common elements) and their union covers $A$.
    2. Partition $\rightarrow$ Equivalence Relation: Given a partition of a set $A$, a relation $R$ defined by $aRb \iff a$ and $b$ belong to the same subset of the partition is an equivalence relation. The resulting equivalence classes are exactly the subsets of the partition.