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370hw0

# 370hw0 - Chapter 0 PRELIMINARIES This chapter contains t h...

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Chapter 0 PRELIMINARIES \ This chapter contains the basic material on sets, functions, relations and induction, as well as the syater of real numbers. Your students may have seen a good share of this material before. You will need to make some choices about what material from this chapter you cover in detail. My experience has been that most students at t h i ~ level have little appreci'ation of the structure of the system of real numbers as a complete ordered f i e l d . For that reason, I have found that a comprehensive coverage of Section 0.5 is very important. oes for Solutions to Exercises 3, The proof is very similar to that of (v) of Theorem 0.2. 4. The proof is very similar to that of (i) of Theorem 0.3. 6, If x E A n B, then x f A and x E 3 , in particular, x E A . Hence, A fl B C A . If x E A , then x E A U B, hence A C A U B. 6 . If x E C\B, then x E C and x f B. Since A C B and x j! B , then x f! A . Therefore, x E C\A. Thus, C\B C C\A. The converse is false as evidenced by the example C = J , A = -1,2,3) and B = (-3,2,3). Here C\A = C\B but A \$ k and B A . 7. A\(A\B) = B if and only i f B C A . To show this, you may want to prove that A\(A\B) = A fl 0. \ 8, L e t x f (A\B) U B \ A ) . Then x f A\l3 or x E B\A. If x E A \ B , then x l A and x f ! B , hence, x E A U B and I x f ! A P 3. If x E B\A, a similar argument shows that % x E A U 0 and x f A fl B. In either case, x E (A U B \ A U B , Now assume x f (A U B)\(A fl 0). Then x E h S B and x f A n B . If x E A , t h e n x \$ B since x f A n B . If xEB, then x f A since x f A fl B. But x E A or x f 3 hence, x E A\B or x E B\A. 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(:\*{, E L u (s\*) = A U B \ A n B . 9 . The point of Russell's paradox is to show the student that using a rule to define a set can lead t o logical d i f f i c u l t i e s . However, reassure your students that no such

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

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