Math 408, Actuarial Statistics I A.J. HildebrandSet-theoretic terminology, and its interpreta-tion in event/outcome languageNotation set-theoretic terminology interpretation(s)S (or Ω) universal set outcome spacea ∈ S element of S individual outcomeA ⊂ S subset of S event (collection of out-comes)A0(or A) complement of A A does not occurthe opposite of A occursA ∪ B union of A and B A or B occurs (non-exclusive “or”)at least one of A and BoccursA ∩ B intersection of A and B both A and B occurA ∩ B = ∅ A and B are disjoint sets A and B are mutually ex-clusiveA and B cannot both occurA ⊂ B A is a subset of B if A occurs then B occursA implies BA \ B set-theoretic difference A occurs, but B does not oc-curSome set-theoretic properties and rules• (A ∪ B)0= A0∩ B0(De Morgan’s Law, I)• (A ∩ B)0= A0∪ B0(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)1Math 408, Actuarial Statistics I A.J. HildebrandNotes and hints• Draw Venn diagrams: Perhaps the most useful piece of advice whenworking with sets is to draw Venn diagrams, especially in more compli-cated situations. Most set-theoretic rules become obvious with a Venndiagram. Instead of memorizing long lists of rules, derive these asneeded through a Venn diagram. The only exceptions are the distribu-tive laws and De Morgan’s laws stated above, which occur frequentlyenough to be worth memorizing. (For practice, try to derive those rulesvia Venn diagrams.)• Use proper set-theoretic notation: Arithmetic operations like ad-dition, subtraction, multiplication don’t make sense in the context ofsets, 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 notationsfor the complement of A: A0,A, Ac, ∼ A. The “dash” notation A0is the one normally used in actuarial exams and in Hogg/Tanis, so wewill stick with this notation.• Complements are rel ative to the underlying universe: In con-trast to other set-theoretic operations (like intersections or union), thedefinition of a complement depends on the “universe” (sample space)under c onsideration. If the underlying “universe” is changed (e.g., re-duced or enlarged), complements change as well. Usually the context ofa problem makes it clear what should be considered as the
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