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Unformatted text preview: 1.1 Variables 5 Example 1.1.4 Rewriting an Existential Universal Statement Fill in the blanks to rewrite the following statement in three different ways: There is a person in my class who is at least as old as every person in my class. Some is at least as old as There is a person p in my class such that p is _. There is a person p in my class with the property that for every person 4 in my class, pis. Solution a. person in my class; every person in my class b. at least as old as every person in my class c. at least as old as q Some of the most important mathematical concepts, such as the definition of limit of a sequence, can only be defined using phrases that are universal, existential, and condi tional, and they require the use of all three phrases &quot;for all,&quot; &quot;there is,&quot; and &quot;ifthen.&quot; For example, if a1 , a2, a3 , . . . is a sequence of real numbers, saying that means that the limit of a, as n approaches infinity is l, for all positive real numbers e , there is an integer N such that forall integers n, if n &gt; N then *t I a,  L &lt; e. Test Yourself , i,3rs to Test Yourselfquestions are located at the end ofeach section. 1 universal statement asserts that a certain property is 3. Given a propefty that may or may not be true, an existential . : _. statement assefis that for which the property is true. ;onditional statement asserts that if one thing _ then re other thing _. *ercise Set 1.1 :r;:lir B contains either full or parlial solutions to all exercises with blue numbers. When the solution is not complete, the exercise ,,':,:: has an l/ next to it. A * next to an exercise number signals that the exercise is more challenging than usual. Be ceLreful not _:. ,.rto the habit of turning to the solutions too quickly. Make every effort to work exercises on your own before checking your .::. See the Preface for additional sources of assistance and further study. a. b. c. T : ,: .ri i6, fill in the blanks using a variable or variables to : r  : :he qiven statement.::re a real number whose square is  1? .. there a real number x such that _? l..es there exist such that x2 : l'l 2. Is there an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when it is divided by 6? a. Is there an integer n such that n has _? b. Does there exist _ such that if n is divided by 5 the remainder is 2 and if ? Note: There are integers with this property. Can you think of one? su ',{le^qrnlur les u Jo &gt;luql uec a,u sesodrnd Ierrteueqluru }soru roC .(S t Ot_Slgt) rolueJ 3roeg,{q 6rgl ur pocnpoJrur s',4a ruJol recrre.'eqle.' IeuroJ e se la.r pro. A or{rJo esn (9861_tggI) 9.{1o4 e8:oag_ .uotssatdxa 1nt4otuat1rutu{o sw.,ro{ aqt qttu.tnuru.nl aq $nu ah ,puuas .uoutpuor ary rQtlnotot1l puD1ilapun $nw att ,tsrtl 'spJot, u, pasodo&quot;rd uo4tpuoc o s10qtu{s l,)t\,aa'pDut ut ,^saldxa o1 \dWail, aA. UaqM . . ....
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This note was uploaded on 06/12/2011 for the course MATH 103 taught by Professor Wouters during the Spring '08 term at Wisc Oshkosh.
 Spring '08
 WOUTERS
 Algebra

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