If a b then a b a union and intersection satisfy

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If A B then A B A . Union and intersection satisfy associative, commutative, and distributive properties. For example, A B C A B A C A B C A B A C 15
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Useful decomposition: If A and B are any subsets of , we can always write A A B A B c , that is, we can write A as the union of two disjoint sets. Also, A B A B A , where A and B A are necessarily disjoint. 16
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DeMorgan’s Laws: A B c A c B c A B c A c B c These are very useful for computing probabilities. 17
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2 . 1 . Infinite Sequences of Sets Let A j : j 1,2,3,. .. be a countable collection of subsets of . Then, by definition, j 1 A j : is in at least one A j j 1 A j : is in every A j , j 1,2,. DeMorgan’s Rules: j 1 A j c j 1 A j c j 1 A j c j 1 A j c 18
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EXAMPLE 5 :( i )L e t A j 1/ j ,1/ j , j 1,2,3,. ... Then j 1 A j 0 . Why? Because zero is the only number in the interval 1/ j j for all j . (ii) Let A j 0,1 1/ j , j 1,2,. j 1 A j . Why? Any number strictly less than one is in a set of the form 1/ j for j large enough. The resulting interval is open on the right because one itself is never in 1/ j for any j . 19
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DEFINITION : (i) A collection of sets A j : j 1,2,. .. is mutually exclusive if A j A h ,all j h . (ii) A collection of subsets A j : j of forms a partition of if the collection is mutually exclusive and  j 1 A j . DEFINITION : (i) A sequence of sets A j : j is said to be monotonically increasing if A j A j 1 for all j 1. (ii) The sequence is monotonically decreasing if A j 1 A j for all j 1. A j 1 1/ j ,1 1/ j for j 1 is monotonically increasing. A j 1/ j ,1/ j is monotonically decreasing. 20
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3 . Sigma Fields When a sample space is discrete, we will take the event space to be the set of all subsets of . This choice simply means that we want to assign a probability to every possible event (even if it may have probability zero). Choosing a proper event space requires more care when is uncountable. While it may seem natural to choose to be the set of all subsets of , this choice is “too large” in a mathematical sense. (We do not have the background to formalize this claim, and it is not very important for our purposes.) 21
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DEFINITION :Let be a nonempty set, and let be a set of subsets of . Then is said to be a field (or algebra )if has the following properties: ( 1) ( 2) If A then A c .( is closed under set complentation .) ( 3) If A 1 , A 2 ,...
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If A B then A B A Union and intersection satisfy...

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