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Math 261/461 Q1 Chapter 3 April 24, 2005

1. In Z+, define U Z+t !e pen if it "ati"fie" the #nditin$ nU e%er& di%i"r f n!eln'" t U. (h) that thi" i" a tpl'& in Z+and that it i" nt the di"#rete tpl'&.

2. *i%e an& definitin r #hara#teriatin f a !a"i" fr a tpl'& and "e it t an")er the

fll)in'$ -eterine )hether r nt the #la"" f pen "are" )ith hrintal and%erti#al "ide" f the plane 2i" a !a"i" fr the "al tpl'& n 2.

3. a -efine a "!!a"i" fr a tpl'&.

! r%e that if Si" a "!!a"i" fr tpl'ie" Tand T*n , then T T*.

4. r%e that if * i" pen in then AGAG = fr e%er&A"!"et f .

5. *i%e an& definitin f the !ndar& rA f A and the interir IntA f A and "e the"e

t pr%e that

a rAIntA ! =A IntA rA6. 7et and * !e den"e in . (h) that if and * are pen, then * i" al" den"e in .

8. A "et i" !i%alent if it i" !th F and a G . (h) that the #pleent", finite nin and

finite inter"e#tin f !i%alent "et" are !i%alent.

9. 7et Z+

)ith the tpl'& f r!le :.1. r%e that if 2121121 nfmfnm

, then++

ZZf $ i" #ntin".

;. 7et YXf $ and ZYg $ . r%e that if fg i" pen andgi" #ntin", in