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. a. Some is at least as old as . .

b. There is a person p in my class such that p is

c. There is a person p in my class with the property that for every person q in my class, p is .

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 conditional, 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 the limit of an as n approaches infinity is L means that for all positive real numbers ε, there is an integer N such that for all integers n, if n > N then −ε < an − L < ε.

Test Yourself
Answers to Test Yourself questions are located at the end of each section. 1. A universal statement asserts that a certain property is for . 2. A conditional statement asserts that if one thing . some other thing then 3. Given a property that may or may not be true, an existential for which the property is true. statement asserts that

Exercise Set 1.1
Appendix B contains either full or partial solutions to all exercises with blue numbers. When the solution is not complete, the exercise number has an H next to it. A ✶ next to an exercise number signals that the exercise is more challenging than usual. Be careful not to get into the habit of turning to the solutions too quickly. Make every effort to work exercises on your own before checking your answers. See the Preface for additional sources of assistance and further study. In each of 1–6, fill in the blanks using a variable or variables to rewrite the given statement. 1. Is there a real number whose square is −1? ? a. Is there a real number x such that such that x 2 = −1? b. Does there exist 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 such that if n is divided by 5 the b. Does there exist ? remainder is 2 and if Note: There are integers with this property. Can you think of one?

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

6 Chapter 1 Speaking Mathematically 3. Given any two real numbers, there is a real number in between. a. Given any two real numbers a and b, there is a real num. ber c such that c is b. For any two , such that a < c < b. d. If J , then J e. For all squares J , . .

4. Given any real number, there is a real number that is greater. s such that s is a. Given any real number r , there is . b. For any , such that s > r . 5. The reciprocal of any positive real number is positive. a. Given any positive real number r , the reciprocal of b. For any real number r , if r is , then . c. If a real number r , then . 6. The cube root of any negative real number is negative. a. Given any negative real number s, the cube root of b. For any real number s, if s is , then . c. If a real number s , then . .

9. For all equations E, if E is quadratic then E has at most two real solutions. . a. All quadratic equations . b. Every quadratic equation . c. If an equation is quadratic, then it , then E . d. If E . e. For all quadratic equations E, 10. Every nonzero real number has a reciprocal. . a. All nonzero real numbers for r . b. For all nonzero real numbers r , there is c. For all nonzero real numbers r , there is a real number s . such that 11. Every positive number has a positive square root. . a. All positive numbers for e. b. For any positive number e, there is c. For all positive numbers e, there is a positive number r . such that 12. There is a real number whose product with every number leaves the number unchanged. has the property that its . a. Some . b. There is a real number r such that the product of r c. There is a real number r with the property that for every . real number s, 13. There is a real number whose product with every real number equals zero. has the property that its . a. Some . b. There is a real number a such that the product of a c. There is a real number a with the property that for every . real number b,

.

7. Rewrite the following statements less formally, without using variables. Determine, as best as you can, whether the statements are true or false. a. There are real numbers u and v with the property that u + v < u − v. b. There is a real number x such that x 2 < x. c. For all positive integers n, n 2 ≥ n. d. For all real numbers a and b, |a + b| ≤ |a| + |b|. In each of 8–13, fill in the blanks to rewrite the given statement. 8. For all objects J , if J is a square then J has four sides. a. All squares . b. Every square . c. If an object is a square, then it .

Answers for Test Yourself
1. true; all elements of a set 2. is true; also has to be true 3. there is at least one thing

1.2 The Language of Sets
. . . when we attempt to express in mathematical symbols a condition proposed in words. First, we must understand thoroughly the condition. Second, we must be familiar with the forms of mathematical expression. —George Polyá (1887–1985)

Use of the word set as a formal mathematical term was introduced in 1879 by Georg Cantor (1845–1918). For most mathematical purposes we can think of a set intuitively, as

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.