RecitationTwoSols

# RecitationTwoSols - Rensselaer Electrical, Computer, and...

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Rensselaer Electrical, Computer, and Systems Engineering Department ECSE 4500 Probability for Engineering Applications Recitation 2 Solutions Wednesday Feb. 3, 2010 Sets and Fields Set Theory De f nition: element arbitrary, can be anything! often denoted a, b, c, . .. De f nition: set a collection of elements, often denoted A, B, C, . .. De f nition: subset A B means that all elements in A are also in B .W e s a y A is a subset of B . De f nition: equality A = B means A B and B A. In words we say A equals B . De f nition: universal set the set containing all the elements De f nition: null set φ the empty set {} De f nition: union A B = { set of elements in either AorB } De f nition: intersection A B = { set of elements in both AandB } De f nition: complement A c = { set of elements in but not in A } De f nition: di f erence A B = { set of elements in A but not in B } Some Basic Set Identities Note: to ease the notational burden, we write A B = AB as is commonly done in this f eld. 1. A A = A 2. AA = A 3. AB = BA 4. A = A 5. A φ = A 6. = φ 7. ( A c ) c = A 8. A A c = 9. A B = B A. To prove this, we must show A B B A and vice versa. To show A B B A we proceed as follows. Let ς be an arbitrary element in A B, i.e. ς A B. Then ς A or ς B. Butthenitmustbethat ς B A by the de f nition of union. To show B A A B, one proceeds analogously. Then having shown both set inclusions A B B A and B A A B, it is true that A B = B A. 10. A c =

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11. AB c = A B. We f rst try to establish that AB c A B. So let ς AB c , it then must be that ς A and ς B c . This is the de f nition of set di
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## This note was uploaded on 04/15/2010 for the course ECSE 4500 taught by Professor Woods during the Spring '08 term at Rensselaer Polytechnic Institute.

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RecitationTwoSols - Rensselaer Electrical, Computer, and...

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