Concepts Basics PDF

## 3.2. CONCEPTOS BÁSICOS ### Definición 3.11 (Partición de Conjuntos) Se denomina partición de un conjunto a su división en subconjuntos mutuamente excluyentes y exhaustivos. ### Ejemplo 3.11 The sets A and B defined in Example 3.10 define a partition of Ω, as A and B are mutually exclusive, A∩B =...

## 3.2. CONCEPTOS BÁSICOS ### Definición 3.11 (Partición de Conjuntos) Se denomina partición de un conjunto a su división en subconjuntos mutuamente excluyentes y exhaustivos. ### Ejemplo 3.11 The sets A and B defined in Example 3.10 define a partition of Ω, as A and B are mutually exclusive, A∩B = ∅, and are exhaustive, AUB = Ω. ### Theorems - **Teorema 3.1 (Idempotencia)** Sea A un subconjunto cualquiera de Ω, entonces - AU A = A - ANA = A - **Teorema 3.2 (Ley Asociativa)** Sean A, B y C, subconjuntos pertenecientes a Ω, entonces - (AUB) UC = AU(BUC) = AUBUC - (A∩B)∩C = A∩(B∩C) = A∩B∩C - **Teorema 3.3 (Ley Conmutativa)** Sean A y B, subconjuntos pertenecientes a Ω, entonces - AUB = B UA - A∩B = B∩A - **Teorema 3.4** Si A y B representan subconjuntos de Ω, entonces - A-B = A∩B^c - **Teorema 3.5 (Identidad)** Sea A un subconjunto cualquiera de Ω, entonces - AU Ω = Ω - ΑΠΩ = A - AU ∅ = A - A∩∅ = Ø - **Teorema 3.6 (Complemento)** Sea A un subconjunto cualquiera de N, then - AU A^c = Ω - A∩A^c = Ø - Ω^c = Ø - ∅^c = Ω - **Teorema 3.7** Sean A y B subconjuntos de Ω, entonces - A = (A∩B) U (A∩B^c) - **Teorema 3.8 (Leyes de Morgan)** Para A y B, subconjuntos cualesquiera de Ω, se cumple que - (AUB)^c = A^c ∩ B^c - (A∩B)^c = A^c U B^c ### **Definition 3.12 (Regla de la multiplicación)** If a procedure can be performed in n₁ different ways and after this, a second procedure can be performed in n₂ different ways, ..., and finally, a k-th procedure can be performed in nk different ways, then the series of k procedures can be performed in n₁ x n₂ x ... x nk different ways. ### Example 3.12 If license plates are made up of two different letters followed by three different digits, how many different license plates can be formed? Given that there are 26 letters and 10 digits, they can be formed in the established way, 26 x 25 x 10 x 9 x 8 = 468000 different license plates. ### Example 3.13 How many lunches consisting of a soup, salad, dessert, and a drink are possible if we can choose from 4 soups, 2 types of sandwiches, 3 desserts, and 2 drinks? 4 x 2 x 3 x 2 = 48 different lunches can be formed. ### Definition 3.13 (Regla de la suma) If a procedure P₁ can be performed in n1 ways and a second procedure P2 can be performed in n2 ways. if they cannot be performed together, then the number of ways that P₁ or P₂ can be performed is n₁ + n₂. ### Example 3.14 To make a trip, you must decide if you will travel by bus or train. If there are three bus routes and two train routes, then there are 4 + 2 = 6 different routes to follow on this trip. ### Definition 3.14 (Permutations) A permutation is an arrangement of all or part of a set of objects in a specific order. Suppose you have n different objects and you want to order r of these objects. There are n ways to choose the first object, n - 1 ways to choose the second object, ... and so on, there are n - r + 1 ways to choose the r-th object. Therefore, the number of arrangements, or different permutations, is given by n(n - 1)(n - 2)... (n - r + 1). This is denoted by nPr and is given by: nPr = n! / (n - r)! ### Example 3.15 If you have a set that consists of n objects, of which n₁ are of one form, n₂ are of a second form, ..., nk are of the k-th form, the number of distinct permutations of the n objects is: n! / n1! n2! ... nk! ### Definition 3.15 (Combinations) It is each of the different arrangements that can be made with part or all of the elements of a set without considering the order in their location. The number of combinations of n different elements taken r at a time, where r ≤ n, is given by: nCr = n! / ((n - r)!r!) ## **3.2.2. Experimentos Aleatorios** It is widely known the importance of experiments in science. A fundamental principle is that if such experiments are carried out repeatedly under approximately identical conditions, the same results are obtained. However, there are experiments where the results are not essentially the same even when repeated under approximately identical conditions, i.e., experiments whose result cannot be predicted before being executed. These experiments are called random experiments. The theory of Probability deals with these experiments. ### Definition 3.16 (Random Experiment) It is any operation whose outcome cannot be predicted with certainty. A random experiment or phenomenon is one that is capable of producing several outcomes, but it is not possible to predict in advance which of them will occur in a particular execution of the experiment. Therefore, a random experiment has the following properties: 1. The experiment can be repeated indefinitely under similar conditions. 2. The set of possible outcomes of the experiment can be known a priori, but a particular outcome cannot be predicted. 3. If the experiment is repeated a large number of times, the proportion with which each outcome appears tends to stabilize, i.e., it tends to a number.