370hw0

370hw0 - Chapter 0 PRELIMINARIES \ This chapter contains t...

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Chapter 0 PRELIMINARIES \ This chapter contains the basic material on sets, functions, relations and induction, as well the syater of real numbers. Your students may have seen a good share this material before. You will need to make some choices about what material from you cover in detail. My experience has been that most students at thi~ level have little appreci'ation structure system real numbers a complete ordered field. For reason, I have found a comprehensive coverage Section 0.5 is very important. oes for Solutions Exercises 3, The proof is very similar to that (v) Theorem 0.2. 4. (i) of Theorem 0.3. 6, If x E An B, then x f A x E particular, x E A. Hence, A fl B C x E A, then x E A U B, hence A C A U B. 6. x E C\B, x E C and x f Since A C B x j! B, x f! Therefore, x E C\A. Thus, C\B C The converse is false as evidenced by example C = J, A = -1,2,3) and B = (-3,2,3). Here C\A = but A $ k and B 7. A\(A\B) = B if only B C To show this, you want prove A\(A\B) = A 0. \ 8, Let x f (A\B) U B\A). Then x f A\l3 or x E B\A. x E A\B, x l A and x B, hence, x E A U B and I x A P 3. If x E B\A, a similar argument shows that % xEAU0 and x f A In either case, x E (A U B \ A U B , Now assume x f U B)\(A 0). Then x E hSB and xfAnB. xEA,then x$B since xfAnB. xEB, xfA since x f A B. But x E A or x f 3 x E A\B x E Thus, x E $A\B) U $B\A~. We have shown that A\B) U B\A) C A U j* v sji[* n sj $ (r\a]'SA(:\*{, EL u (s\*) = AUB\AnB. 9. point Russell's paradox is to show student using a rule to define a set can lead logical difficulties. However, reassure your that no such
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problems will arise in this book. 11, This proof is very similar to that of (i) Theorem 0.4. One distinction is if x E S\( fl A1), then x E S 1 €A and x 6 AX, hence x A for some p E A. Therefore, AEA P x E S\Ap, etc., etc. 12. a) R\{o} *b) R\C1,2] 13. im f = {m E J: a odd). f 1 - 1, but not onto J.
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This note was uploaded on 04/02/2009 for the course MAT 370 taught by Professor Kuiper during the Spring '09 term at ASU.

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370hw0 - Chapter 0 PRELIMINARIES \ This chapter contains t...

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