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Session_2 1-14-10

# Session_2 1-14-10 - Set Theory 8{1 32;3 g4g5g 6...

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Unformatted text preview: Set Theory 8 : {1 32 ;3 g4g5g 6; ﬁlaamgglé gfﬁgg We can deﬁne events A1 = outcome is odd 2 {1.3.53 A2 : outcome is divisible by 3 :{3 , 63 A3 : outcome is prime =12 , 3' 5 S Note: Each event of interest is described by a subset of S : {1,2,3,4,5,6§ There are 26 = 64 distinct subsets of S. In order to fully characterize this random experiment, we must know the probability of each of these sets. Events: Aka 5 HS) :{A 1‘ 142‘ ' A64; f the set of all events. ‘3 Q V06? sf‘vce. Our random experiment is completely characterized by where and assigns probability to each of the events. This framework—with minor modiﬁcations—will be used to describe all of the random experiments we will conﬂden A solid understanding of set theory is essential to this task. Basic Set Theory Deﬁnitions 1, A set is simply a collection of objects rewe intentionally leave this undeﬁned. Defn: In any given problem, the set containing all possible elements of interest is called the universe, universal set, or Space. We typically denote the space by 8 0c There are a number of basic set Operations we must be familiar with: Defn: The union of two sets A and B7 denoted A U B, is deﬁned as A 0 B m AUB:{wES:wererB}. Defn: The intersection of two sets A and B, denoted A F] B. is deﬁned as A “9 “ AﬂB: {w€8: wEAandwEB} wéA an! "’53 Defn: The complement of a set A (with respect to S), denoted A A , or AC is deﬁned # as {wESz ng4} K— {we}: WéA Z Basic Set Theory Deﬁnitions Leont.) Defn. The set containing no elements is called the empty set or null set, and is denoted by 0 or {}. (n.b. {(0} is not correct notation.) { 3 O ' Defn: If two sets A and B have no elements in common, then A H B 2 Q), and A and B are said to be disjoint. —-- AAE s ¢ 0 The set differenceof two sets A and B is deﬁned as eB:{wES:wEAandw¢B}. A —B ={1wei :weAaAwaj:Al\§ Defn: The symmetric difference of two sets A and B , is defined as M AAB={wESszAorwgéB,butnotboth.}. m A AB = (Aum- mm = (Auem (AAB) Defn: Two sets A and B are equal if they contain exactly the same elements. — Fact: Two sets A and B are equal if and only if (iﬂ?) A C B and B C A —_— —-I—- Proof: exercise. A .3 B a A CB an." RCA '==7 g Algebra of Set Theory 1. AJBzBUA '5. c.ww.w‘4‘h¢¢ (Ra’s W" 2.AnB=BnA “Jute " 3. AJ(BUC)=(AUB)UC. 1), A99" 4.AW(BﬂC)=(AnB)ﬂC. uro- 5.Aw(Bo0):(AnB)o(AnC) 95+..u‘“ '3 6.AJ(BﬂC):(AUB)ﬂ(AUC) 7.2: L \$ 8.A’1B=AUB C ““5 1.01 9.AJB:AOB} DC M07 10.320 ‘) 11 A’WS:A 12l\$ ;- An£=hn(BU%AAB—) 13.AJS=S 14.AJ(D:A =®AB)U 15.AJA:S l6.A*7A=(D L Lsd (521': “My and}. TknJ (, Mr A; ' , ﬂ“ , {Ruthie-clad “4'0”“ {3 / #7 m c4; “.14 ﬁﬂtt" a; 52"} iA‘3LGI; - 4M Vtvuovt ~- 9F ~r-Lt'5 4;“, 'c: U A = {noel 006A 42: .+l«.,t Fﬂf‘tt‘ton a; «8. umJ leé I, I. 31.9,) (A “”919.“ .,c 3 7“: f'oLQLc’v’IU F(A\ r19! ...
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