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homework1 - 1.1 Variables 5 Example 1.1.4 Rewriting an...

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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 "for all," "there is," and "if-then." 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 > N then *t I a, - L < 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 i-6, 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?
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