Math 408 Actuarial Statistics I A J Hildebrand Set theoretic terminology and its interpretation in event outcome language Notation set theoretic terminology interpretation s S or universal set outcome space a S element of S individual outcome A S subset of S event collection of outcomes A0 or A complement of A A does not occur the opposite of A occurs A B union of A and B A or B occurs nonexclusive or at least one of A and B occurs A B intersection of A and B both A and B occur A B A and B are disjoint sets A and B are mutually exclusive A and B cannot both occur A B A is a subset of B if A occurs then B occurs A implies B A B set theoretic difference A occurs but B does not occur Some set theoretic properties and rules A B 0 A0 B 0 De Morgan s Law I A B 0 A0 B 0 De Morgan s Law II A B C A B A C Distributive Law I A B C A B A C Distributive Law II 1 Math 408 Actuarial Statistics I A J Hildebrand Notes and hints Draw Venn diagrams Perhaps the most useful piece of advice when working with sets is to draw Venn diagrams especially in more complicated situations Most set theoretic rules become obvious with a Venn diagram Instead of memorizing long lists of rules derive these as needed through a Venn diagram The only exceptions are the distributive laws and De Morgan s laws stated above which occur frequently enough to be worth memorizing For practice try to derive those rules via Venn diagrams Use proper set theoretic notation Arithmetic operations like addition subtraction multiplication don t make sense in the context of sets and you shouldn t use arithmetic notations like with sets Thus write A B not A B A B not A B etc Notations for complements There are several common notations for the complement of A A0 A Ac A The dash notation A0 is the one normally used in actuarial exams and in Hogg Tanis so we will stick with this notation Complements are relative to the underlying universe In contrast to other set theoretic operations like intersections or union the definition of a complement depends on the universe sample space under consideration If the underlying universe is changed e g reduced or enlarged complements change as well Usually the context of a problem makes it clear what should be considered as the underlying universe 2
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