Introduction to Probability: Problem Solutions
Introduction to Probability: Problem Solutions (last updated: 5/15/07)c Dimitri P. Bertsekas and John N. TsitsiklisMassachusetts Institute of TechnologyWWW site for book information and Scientific, Belmont, Massachusetts1C H A P T E R 1Solution to Problem haveA={2,4,6},B={4,5,6},soA B={2,4,5,6}, and(A B)c={1,3}.On the other hand,Ac Bc={1,3,5} {1,2,3}={1,3}.Similarly, we haveA B={4,6}, and(A B)c={1,2,3,5}.On the other hand,Ac Bc={1,3,5} {1,2,3}={1,2,3,5}.Solution to Problem (a) By using a Venn diagram it can be seen that for anysetsSandT, we haveS= (S T) (S Tc).(Alternatively, argue that anyxmust belong to eitherTor toTc, soxbelongs toSif and only if it belongs toS Tor toS Tc.) Apply this equality withS=AcandT=B, to obtain the first relationAc= (Ac B) (Ac Bc).Interchange the roles ofAandBto obtain the second relation.(b) By De Morgan s law, we have(A B)c=Ac Bc,and by using the equalities of part (a), we obtain(A B)c=((Ac B) (Ac Bc)) ((A Bc) (Ac Bc))= (Ac B) (Ac Bc) (A Bc).
Introduction to Probability: Problem Solutions (last updated: 5/15/07) c Dimitri P. Bertsekas and John N. Tsitsiklis Massachusetts Institute of Technology WWW site for book information and orders
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