Introduction to Set Theory

Introduction to Set Theory - Introduction to Set Theory...

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Unformatted text preview: Introduction to Set Theory (and brief discussion of probability) Population‐ the collection of all individuals or items under consideration In this course we denote population by Ω (the capital letter omega). Example: If we roll a die one time, then Ω ={1,2,3,4,5,6}. Ω represents all POSSIBLE OUTCOMES from the “random experiment” or the model under consideration. Loosely speaking, probability is a percentage. More specifically, we have the following concept: Suppose that a member is selected at random from a finite population. Then the probability that the member obtained has a specified attribute equals the percentage of the population that has that attribute. Example: Suppose a Stat 225 class consists of 30 males and 10 females. What is the probability that a randomly selected student [from this particular Stat 225 class] is male? Random Experiment‐ an action whose outcome cannot be predicted with certainty (there is some randomness involved in the action). Event‐ some SPECIFIED result that may or may not occur when the random experiment is performed. Remember, an event can be simple (only one specified result) or it can be a collection of results. (The class example of rolling a die: A‐5 is rolled, B‐ 3 is rolled, C‐ an odd is rolled. A and B are simple events but C is not). Frequentist Interpretation of Probability The probability of an event is the long‐run proportion of times that the event occurs in independent repetitions of the random experiment. (Independence is a key topic in probability.) This can be written as: p ( E ) = n( E ) . The n represents the sample size. p(E) is the probability of event E n occurring (p always represents probability; when clarification is not needed we will just use p). n(E) is the number of times event E occurs in the n repetitions. LONG‐RUN means that n is LARGE. There are differing viewpoints on large (typical examples are >100, >1,000, >1,000,000 etc.). We will not use this exact formula. We will use the concept though. For something that is THEORETICALLY TRUE (not just in the long‐run) we have the following: p( E ) = number of possible ways E can occur . n (Ω ) In essence, our probability is the number of ways to get what we want (are looking for) divided by the total number of possibilities. Caution: This is true for very basic set‐ups: namely, we have all outcomes are equally likely and (if applicable) independent repetitions. Our typical examples are rolling a fair die and flipping a fair coin. This is no longer true when outcomes are NOT equally likely. Set Theory Set‐ a collection of elements (numbers, items, etc.). Let A be a set. Let x be an element. If x is in A (belongs to) then we write x ∈ A . If x is not in A, we write x ∉ A . ∅ represents the empty set (the set with no elements). Ω represents the population, or the set of all possible outcomes. Ω and ∅ are complements. A complement is a set that contains all of the elements in the population that are not in the original set. We denote a complement with a superscript C. Ex (1 die roll): If A is {2,4,6} then AC is {1,3,5}. Subsets‐ Let A and B be sets. If every element of A is an element of B, then A is a subset of B. This is denoted A ⊂ B or B ⊃ A . Typically we will use the first expression. Note that the set that is possibly larger has the open end of the symbol signifying a “spreading out”. I say possibly larger becauseB may or may not contain additional elements (elements other than the ones that appear in A). If B does not contain additional elements then the 2 sets are equal. If B does contain additional elements then A is a proper subset of B. It is typical when talking about subsets to interpret the symbol as “is contained in”. IMPORTANT SETS: R = collection of real numbers Z=collection of integers Q=collection of rational numbers (can be viewed as a fraction) N=collection of positive integers Intervals of real numbers: Let a and b be real numbers (here these are our end points): (a,b)={x ∈ R: a < x < b} (the colon is usually read as such that) [a,b]={x ∈ R: a ≤ x ≤ b} Note the difference between () and . () are called open intervals whereas are called closed intervals. If your interval contains infinity then it is considered unbounded (no infinity implies bounded). Also, conventionally, if you use ∞ or -∞ as one of your end points, then you use a ( with it. Ex: ( -∞ , 5] or ( -∞ ,5) depending on whether you want to include 5. IMPORTANT DEFINITIONS Let A, B, and C be subsets of Ω . 1) The complement of A is AC = {x : x ∉ A} . It is all the elements in omega that are not in A. Ex.: A={1,6} and Ω ={1,2,3,4,5,6} then AC is {2,3,4,5}. 2) The intersection of A and B is A ∩ B = { x : x ∈ A and x ∈ B} . Intersection means it is in both (all if more than 2 sets). Intersection implies “and”. Let A={22,42,62,82} and B={‐40,‐20,0,20} then their intersection is ∅ since they have no elements in common. 3) The union of A and B is A ∪ B={x:x ∈ A or x ∈ B} . Union means it is in one or the other (at least one of the sets). Note: We do not double count. Ex: Let A={1,2,3} and B={2,4,6} then their union is {1,2,3,4,6} NOT {1,2,2,3,4,6}. De Morgan’s Laws Let A and B be subsets of Ω . 1) (A ∪ B)C = AC ∩ B C 2) ( A ∩ B )C = AC ∪ B C Proof of part 1 to be done IN CLASS (for equality must prove containment both ways). Venn Diagrams are useful for helping you decide if a statement is true or false, but it is NOT A PROOF. Distributive Laws: Let A, B, and C be subsets of Ω . 1) A ∩ ( B ∪ C ) = ( A ∩ B) ∪ ( A ∩ C ) 2) A ∪ ( B ∩ C ) = ( A ∪ B) ∩ ( A ∪ C ) Associative and Commutative Laws Let A, B, and C be subsets of Ω . 1) A∩ B = B ∩ A 2) A∪ B = B ∪ A 3) A ∩ ( B ∩ C ) = ( A ∩ B) ∩ C 4) A ∪ ( B ∪ C ) = ( A ∪ B) ∪ C Intersections and Unions (We can perform these operations on more than just 2 sets) Let A1 , A 2 , ... be a countably infinite collection of subsets of Ω . (We can shorten this to a finite set if we wish.) ∞ I A = {x : x ∈ A i i for all i} i =1 ∞ U Ai = {x : x ∈ Ai for at least one i} i =1 De Morgan’s Laws (new) Let A1 , A 2 , ... be a countably infinite collection of subsets of Ω . 1) C (I An )C = U An n for any positive integer n. for any positive integer n. n C ⎛ ⎞ C 2) ⎜ U An ⎟ = I An n ⎝n ⎠ Distributive Laws (new) Let B, A1 , A 2 , ... be a countably infinite collection of subsets of Ω . 1) ⎛ ⎞ B ∩ ⎜ U An ⎟ = U ( B ∩ An ) ⎝n ⎠n 2) B ∪ (I An ) = I ( B ∪ An ) n n Do not feel overwhelmed by these formulas. They are merely an extension of the 2 set formulas to 3 or more sets. In practice, we will not work with a large number of sets. Disjoint Sets‐ Two sets A and B are said to be disjoint sets if A ∩ B = ∅ . This states that their intersection is the empty set, or that they have no elements in common. Pairwise Disjoint‐ For sets A1 , A 2 , ... , they are said to be pairwise disjoint if Ai ∩ A j = ∅ whenever i ≠ j. This implies that every pair is disjoint. ...
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