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Unformatted text preview: 2392.4 Sequences and Summations38d)xy=xyfor all real numbersxandy.e)x2=x+12for all real numbersx.70.Prove or disprove each of these statements about the floorand ceiling functions.a)x=xfor all real numbersx.b)x+y=x+yfor all real numbersxandy.c)x/2/2=x/4for all real numbersx.d)√x=√xfor all positive real numbersx.e)x+y+x+y≤2x+2yforallrealnumbersxandy.71.Prove that ifxis a positive real number, thena)√x=√x.b)√x=√x.72.Letxbe a real number. Show that3x=x+x+13+x+23.A program designed to evaluate a function may not producethe correct value of the function for all elements in the domainof this function. For example, a program may not produce acorrect value because evaluating the function may lead to aninfinite loop or an overflow. Similarly, in abstract mathematics, we often want to discuss functions that are defined only fora subset of the real numbers, such as 1/x,√x, and arcsin (x).Also, we may want to use such notions as the “youngest child”function, which is undefined for a couple having no children,or the “time of sunrise,” which is undefined for some daysabove the Arctic Circle.To study such situations, we use the concept of a partialfunction. Apartial functionffrom a setAto a setBis anassignment to each elementain a subset ofA, called thedomain of definitionoff, of a unique elementbinB. The setsAandBare called thedomainandcodomainoff, respectively. We say thatfisundefinedfor elements inAthat arenot in the domain of definition off. We writef:A→Btodenote thatfis a partial function fromAtoB. (This is thesame notation as is used for functions. The context in whichthe notation is used determines whetherfis a partial functionor a total function.) When the domain of definition offequalsA, we say thatfis atotal function.73.For each of these partial functions, determine its domain,codomain, domain of definition, and the set of values forwhich it is undefined. Also, determine whether it is a totalfunction.a)f:Z→R,f(n)=1/nb)f:Z→Z,f(n)=n/2c)f:Z×Z→Q,f(m,n)=m/nd)f:Z×Z→Z,f(m,n)=mne)f:Z×Z→Z,f(m,n)=m−nifm>n74. a)Show that a partial function fromAtoBcan be viewedas a functionf∗fromAtoB∪ {u}, whereuis not anelement ofBandf∗(a)=⎧⎨⎩f(a)ifabelongs to the domainof definition offuiffis undefined ata.b)Using the construction in (a), find the functionf∗corresponding to each partial function in Exercise 73.75.a)Show that if a setShas cardinalitym, wheremis apositive integer, then there is a onetoone correspondence betweenSand the set{1,2,...,m}.b)Show that ifSandTare two sets each withmelements, wheremis a positive integer, then there is aonetoone correspondence betweenSandT.76.∗Show that a setSis infinite if and only if there is a propersubsetAofSsuch that there is a onetoone correspondence betweenAandS....
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This note was uploaded on 12/16/2009 for the course MATH 1014 taught by Professor Ganong during the Spring '09 term at York University.
 Spring '09
 ganong
 Real Numbers

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