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Unformatted text preview: ¯ ELEMENTARY PROBABILITY ¯ ¯ AND STOCHASTIC PROCESSES ¯ Nguyen Van Thu Professor Dr. HCMIU VN 1 SET THEORY This Section treats some elementary ideas and concepts of set theory which are nec essary for a modern introduction to probability theory. Any well defined list or collection of objects is called its elements or members . We write p ∈ A , if p is an element in the set A . We say that A is a subset of B, if every element of A belongs to set B. This is denoted by A ⊂ B Two sets are equal if each is contained in the other, that is A = B if and only if A ⊂ B and B ⊂ A We specify a particular set either by listing its elements or by stating properties which characterize the elements of the set. For example, A = { 1 , 3 , 5 , 7 , 9 } B = { x : x is a prime number ,x < 15 } . All sets under investigation are assumed to be subsets of some fixed set called the universal set and denoted by Ω. We use ∅ to denote the empty or null set i.e the set which contains no elements; this set is regarded as a subset of every other set. Ex: N = the set of positive integers : 1,2,3,... Z = the set of integers: ··· ,2,1,0,1,2,3, ··· Thus we have N ⊂ Z ⊂ R Theorem 1.1 Let A,B and C be any sets. Then (i) A ⊂ A ;(ii) if A ⊂ B and B ⊂ A then A = B ; and (iii) if A ⊂ B and B ⊂ C then A ⊂ C A. SET OPERATIONS 1 Let A and B be arbitrary sets. The union of A and B , denoted by A ∪ B is the set of elements which belong to A or B : A ∩ B = { x : x ∈ A or x ∈ B } . The intersection of A and B , denoted by A ∩ B , is the set of elements which belong both to A and B : A ∩ B = { x : x ∈ A and x ∈ B } . If A ∩ B = ∅ then A and B are said disjoint . The difference of A and B or the relative complement of B with respect to A, denoted by A \ B , is the set of elements which belong to A but not to B : A \ B = { x : x ∈ A,x / ∈ B } . Observe that A \ B and B are disjoint, i.e ( A \ B ) ∩ B = ∅ . The absolute complement or complement of A , denoted by A c , is the set of elements which do not belong to A : A c = { x : x ∈ Ω ,x / ∈ A } . That is, A c is the difference of the universal set Ω and A . Venn diagrams B. FINITE AND COUNTABLE SETS A set is finite if it is empty or if it consists of exactly n elements where n is a positive integer, otherwise it is infinite. EX: Let M be the set of the days of week. M = { Monday,Tuesday,..., Sunday } . Then M is finite EX: Let Y be the set of even integers Y = { 2 , 4 , 6 ···} . Then Y is an infinite set. A set is countable if it is finite of if its elements can be arranged in the form of a sequence. 2 Two examples above are countable sets. EX: Let I be the unit interval of real numbers; i.e I = { x : 0 6 x 6 1 } ....
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 Spring '09
 thu
 Set Theory, Probability

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