e1key - M413 EXAM I Name J:;~-~ There should be 5 distinct...

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M413 EXAM I Name J:;~-~ There should be 5 distinct pages with a total of 9 problems. Consider the following sets JR set of real numbers, Q Z = set of integers, and anxn + . ..+ alX + aO ,an,bo, bm E JR, bm =I 0 lao, ~ = ~ f(x) = -- l bmxm + . ..+ bI x + bo with the usual arithmetic, along with 1 ,I. JF={O,1,2},.arithmeticmOd.3. .,15 pts.) LIst all the systems above whIch fulfill the axIOms of the folloWIng ~ tZ 7i 1R It: (a) ring (b) field FF ~ R 11< IR (£( R ( c) ordered field ~ rQ (d) Archimedean ordered field fR. ( e) complete ordered field 2. (15 pts.) For the system types (b) through (e) in problem 1, state the axioms that each must e~oy, that the system before it does NOT. (b) "Z'~;:- ~'?G "V-"'/~ Dr1-I(\.J: ~ \ s.f: /'x-:-X \.7ix' I .'X-fo I '3 ~-I s.t 'X-~)'-= I 041 ~r' el\ '=J ~ S.-t. (c) X.!x \Lx )<~::))~f".=:; K-=--'1 X~~) ~t-z. =:) Xf, ~ I=D~ ~",u,;,. . x"d t\~ x~ 'J ~ -) ~ x+z ~ ~-I--2. () ~A 0 L \A ~ ) -(J -j D £ )<.~ ~or ~. ')(. 3 ;.Jf: 7L.Lst . ~~)<. "r (d) ?G~ 2' '10 J.J. se (e) t'-\~~ : c(,.V\\Je. .r~ l V\ <J\. . ft;"~ 1
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3 6. (10 pts. Consider the sets B = B([O, 1.]) = {f : [0,1] -+ IRlf is bounded} , C = C([O, 1]) = {f : [0,1] -+ IRlf is continuous} , n V = {X E ]Rn 1 ~ x~ ~ I} k=l with the usual vector arithmetic. Consider also the norms and metrics
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This note was uploaded on 01/26/2010 for the course MATH-M 413 taught by Professor Michaeljolly during the Spring '08 term at Indiana.

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e1key - M413 EXAM I Name J:;~-~ There should be 5 distinct...

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