Logical Symbols Table

Lógica: Tabela de Símbolos Lógicos

• $\land$: Conjunção, significa "e".
• $\lor$: Disjunção, significa "ou".
• $\lnot$: Negação, significa "não".
• $\rightarrow$: Condicional, significa "se... então".
• $\leftrightarrow$: Bicondicional, significa "se e somente se".
• $\oplus$: Disjunção Exclusiva, significa "ou... ou".
• $\therefore$: Conclusão, significa "portanto".
• $\Leftrightarrow$: Equivalência Lógica, significa "é logicamente equivalente a".
• $\Rightarrow$: Implicação Lógica, significa "implica logicamente em".
• $\forall$: Quantificador Universal, significa "para todo".
• $\exists$: Quantificador Existencial, significa "existe".
• $\nexists$: Negação do Existencial, significa "não existe".
• $\exists!$: Existência e Unicidade, significa "existe um único".
• $\in$: Pertinência, significa "pertence a".
• $\notin$: Não Pertinência, significa "não pertence a".
• $\subset$: Inclusão, significa "está contido em".
• $\not\subset$: Não Inclusão, significa "não está contido em".
• $\subseteq$: Inclusão Ampla, significa "está contido ou é igual a".
• $\not\subseteq$: Não Inclusão Ampla, significa "não está contido nem é igual a".
• $\supset$: Contém, significa "contém".
• $\not\supset$: Não Contém, significa "não contém".
• $\supseteq$: Contém Ampla, significa "contém ou é igual a".
• $\not\supseteq$: Não Contém Ampla, significa "não contém nem é igual a".
• {}: Conjunto, significa "o conjunto".
• $\varnothing$: Conjunto Vazio, significa "conjunto vazio".
• $\mathbb{N}$: Naturais, significa "o conjunto dos números naturais".
• $\mathbb{Z}$: Inteiros, significa "o conjunto dos números inteiros".
• $\mathbb{Q}$: Racionais, significa "o conjunto dos números racionais".
• $\mathbb{R}$: Reais, significa "o conjunto dos números reais".
• $\mathbb{C}$: Complexos, significa "o conjunto dos números complexos".
• $\mathbb{P}$: Primos, significa "o conjunto dos números primos".
• $|x|$: Valor Absoluto, significa "o valor absoluto de x".
• $n!$: Fatorial, significa "n fatorial".
• $\sum_{i=1}^{n} x_i$: Somatório, significa "somatório de $x_i$ de 1 até n".
• $\prod_{i=1}^{n} x_i$: Produtório, significa "produtório de $x_i$ de 1 até n".
• $\lim_{x \to a} f(x)$: Limite, significa "o limite de f(x) quando x se aproxima de a".
• $\frac{dy}{dx}$: Derivada, significa "a derivada de y em relação a x".
• $\int f(x) , dx$: Integral, significa "a integral de f(x) em relação a x".
• $\infty$: Infinito, significa "infinito".
• $\neg p$: Negação de p, significa "não p".
• $p \land q$: Conjunção de p e q, significa "p e q".
• $p \lor q$: Disjunção de p e q, significa "p ou q".
• $p \rightarrow q$: Condicional de p e q, significa "Se p, então q".
• $p \leftrightarrow q$: Bicondicional de p e q, significa "p se e somente se q".

Lógica: Falácias Lógicas

• Ad hominem
• Apelo à autoridade
• Espantalho
• Falso dilema
• Derrapagem
• Inversão do ónus da prova
• Apelo à emoção
• Banda
• Generalização apressada
• Falsa causa
• Falácia da composição
• Falácia da divisão

Fonction Logarithme Népérien: Définition

• La fonction logarithme népérien (ln) est définie sur $]0; +\infty[$.
• C'est la primitive de $x \mapsto \frac{1}{x}$ qui s'annule en 1.
• $ln(x)$ est dérivable sur $]0; +\infty[$, et $(ln(x))' = \frac{1}{x}$.
• $ln(x)$ est continue sur $]0; +\infty[$.
• $ln(1) = 0$.

Fonction Logarithme Népérien: Propriétés Algébriques

• Pour $a, b > 0$ et $n \in \mathbb{Z}$:
  • $ln(ab) = ln(a) + ln(b)$
  • $ln(\frac{1}{a}) = -ln(a)$
  • $ln(\frac{a}{b}) = ln(a) - ln(b)$
  • $ln(a^n) = n \cdot ln(a)$
  • $ln(\sqrt{a}) = \frac{1}{2}ln(a)$

Fonction Logarithme Népérien: Étude

• La fonction ln est strictement croissante sur $]0; +\infty[$.
• $\lim_{x \to +\infty} ln(x) = +\infty$
• $\lim_{x \to 0} ln(x) = -\infty$
• $\lim_{h \to 0} \frac{ln(a+h) - ln(a)}{h} = \frac{1}{a}$ pour $a > 0$.
• $\lim_{x \to 0} \frac{ln(1+x)}{x} = 1$
• $\lim_{x \to +\infty} \frac{ln(x)}{x} = 0$
• $\lim_{x \to +\infty} \frac{ln(x)}{x^n} = 0$ pour $n \in \mathbb{N}^*$.
• $\lim_{x \to 0} x \cdot ln(x) = 0$

Fonction Logarithme Népérien: $x \mapsto ln(u(x))$

• Si u est strictement positive et dérivable sur I, alors $x \mapsto ln(u(x))$ est dérivable sur I.
• $(ln(u(x)))' = \frac{u'(x)}{u(x)}$
• $f(x) = ln(2x + 3)$ est définie et dérivable sur $]-\frac{3}{2}; +\infty[$, et $f'(x) = \frac{2}{2x+3}$.
• $g(x) = ln(x^2 + 1)$ est définie et dérivable sur $\mathbb{R}$, et $g'(x) = \frac{2x}{x^2+1}$.

Static Electricity: Charging by Friction

• Involves the triboelectric series, which lists materials by their tendency to become positively or negatively charged when rubbed together.
• Materials higher on the series become more positive when rubbed with those below.
• Electrons are transferred between objects; one becomes positive, the other negative.
• Best results occur with insulators. Also known as triboelectric charging.
• Charge depends on a material's affinity for electrons.
• Glass becomes positive and silk negative when rubbed together.

Static Electricity: Charging by Conduction

• A charged object directly contacts a neutral object.
• Electrons transfer between objects until they have the same charge.
• Requires contact; works best with conductors.
• A negatively charged rod gives electrons to a neutral metal sphere, charging it negatively.

Static Electricity: Charging by Induction

• A charged object is brought near a neutral object (no contact).
• Polarization occurs: electrons in the neutral object shift.
• The neutral object is grounded allowing electrons to flow in or out.
• Ground is removed, resulting in the neutral object having the opposite charge of the charged object.
• A negatively charged rod near a neutral sphere repels electrons, grounding removes excess electrons, and the sphere becomes positively charged.

Teorema de Green: Generalidades

• Relaciona una integral de línea alrededor de una curva plana simple $C$ con una integral doble sobre la región $D$ limitada por $C$.
• Es la versión bidimensional del teorema de Stokes.

Teorema de Green: Enunciado Formal

• Sea $C$ una curva plana, cerrada, simple y positivamente orientada, diferenciable a trozos, y sea $D$ la región limitada por $C$.
• Si $P$ y $Q$ tienen derivadas parciales continuas en una región abierta que contiene a $D$, entonces $\oint_C P dx + Q dy = \iint_D (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) dA$
• La integral de línea se recorre en sentido antihorario alrededor de $C$.

Teorema de Green: Demostración

• $D$ se puede expresar como región de tipo I: $D = {(x, y) | a \le x \le b, g_1(x) \le y \le g_2(x) }$.
• $D$ se puede expresar como región de tipo II: $D = {(x, y) | c \le y \le d, h_1(y) \le x \le h_2(y) }$.
• $\iint_D (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) dA = \iint_D \frac{\partial Q}{\partial x} dA - \iint_D \frac{\partial P}{\partial y} dA$.
• $\iint_D \frac{\partial P}{\partial y} dA = - \oint_C P(x, y) dx$.
• $\iint_D \frac{\partial Q}{\partial x} dA = \oint_C Q(x, y) dy$.

Chemical Kinetics: Reaction Rate

• For the general reaction $ aA + bB \rightarrow cC + dD $
• The rate can be expressed as: $Rate = -\frac{1}{a} \frac{d[A]}{dt} = -\frac{1}{b} \frac{d[B]}{dt} = \frac{1}{c} \frac{d[C]}{dt} = \frac{1}{d} \frac{d[D]}{dt} $

Chemical Kinetics: Rate Law

• For the general reaction $ aA + bB \rightarrow cC + dD $
• The rate law can be expressed as: $Rate = k[A]^x [B]^y$
  • k is the rate constant
  • x is the order with respect to A
  • y is the order with respect to B
  • x+y is the overall order of reaction

Chemical Kinetics: Integrated Rate Law

• Zero Order: $A \rightarrow products$

  • Rate = k
  • $[A]_t = -kt + [A]_0$
  • $t_{1/2} = \frac{[A]_0}{2k}$

• First Order: $A \rightarrow products$

  • Rate = k[A]
  • $ln[A]_t = -kt + ln[A]_0$
  • $t_{1/2} = \frac{0.693}{k}$

• Second Order: $A \rightarrow products$

  • Rate = k[A]^2
  • $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$
  • $t_{1/2} = \frac{1}{k[A]_0}$

Chemical Kinetics: Half-Life

• Time required for the reactant concentration to decrease to one half of its initial value.

Chemical Kinetics: Arrhenius Equation

• Arrhenius Equation: $k = Ae^{\frac{-E_a}{RT}}$
  • $E_a$ is the activation energy
  • R is the gas constant (8.314 J/mol·K)
  • A is the frequency factor
  • $ln(k) = \frac{-E_a}{R} (\frac{1}{T}) + ln(A)$

Chemical Kinetics: Catalysis

• Catalyst: substance that increases the rate of a chemical reaction without being consumed.
• Homogeneous catalyst: present in the same phase as the reactants.
• Heterogeneous catalyst: present in a different phase than the reactants.

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