Modern Elementary Mathematics 0155610430

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MOD6RN €L€M€NTfiRV MATH€MflTICS Fourth €dition

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m

List of Symbols Symbol P^Q

Meaning If P. then £> (or P implies 0) The set whose elements are a, b. c, and so on

[a,b, c...} xEA

x is an element of set A

x£A

x is not an element of set A Such that: divides

=

Is equal to

f 0or{

Is not equal to }

ACB

The empty set A is a subset of B

ACB

A is a proper subset of B

i ADB

The complement of set A The intersection of sets A and B

AUB

The union of sets A and B

{a, b) Ax B

Ordered pair The Cartesian product of sets A and B

1-1

One-to-one correspondence

A-B n(A)

Set A is equivalent to set B The cardinal number of set A

\ = {\. 2. 3....}

The set of natural numbers

W'={0. 1. 2. 3,...}

The set of whole numbers

a - b

The sum of a and b

ab. a x b. (

Is less than or equal to Is greater than

* > J u

« (,( 1)

Is not greater than [s greater than or equal to The set of integers The additive inverse of a Absolute value of the number i Greatest common dnisor

^xi

Symbol

Meaning

I( M

Least common

[a]

The equivalence class of a

Q

The set of rational numbers

x ' R

The multiplicative inverse of x The set of real numbers

—

Is approximately equal to Percent

i AB

The positive square root of x Line AB

AB

Line segment AB

AB

Open line segment AB

AB

Half-open line segment AB

AB

Half-line AB

A~B tABC

Ray AB Angle ABC

multiple

Is congruent to (for modular systems)

Is congruent to (for geometric figures) Is parallel to 1

Is perpendicular to

"i (.-A )

The measure of set A

LABC

The triangle with vertices A. B. and C

a

The standard deviation of a set of numbers

Pi A)

The probability of A

P(A )

The probability of not-A

n! P(n.r)

n\ = n (n - 1) (n - 2) ■■■ (2)(1) The number of permutations of n elements taken rata time

C(n.r)

The number of combinations of n elements taken r at a time

Is similar to (for geometric figures)

P\J Q

Statement P or statement Q

P A Q

Statement P and statement Q

P - 1 with k less than 5 and k G W}.

SOLUTION:

The values of k such that k is less than 5 and k E W are 0. 1 . 2 ; . and 4. Letting * = 0, 1.2. 3. and 4. we obtain S = {1. 3. 5. 7, 9}.

A GCAMPIC 2 SOLUTION:

Use the roster method to indicate {y|y2 = 81 and >£/}. Since (~9)2 = 81 and 92 = 81. the given set can be indicated in roster form as {_9,9}.

We emphasize that there is no unique correct method for indicating a set using set-builder notation: several different and perhaps equally useful methods for identifying the elements of any given set may be formulated. However, a set should be defined carefully and clearly so that it is possible to tell whether or not any given element is a member of that set. Such a set is said to be well-defined. A €XRMPL€ 3

Each of the following sets is well-defined:

(a) The set of past Presidents of the United States of America. (b) {x\x is a whole number less than 15}. (c) The 1983 graduating class of the University of Nevada. Las Vegas. The (a) (b) (c)

following sets are not well-defined: The set of all intelligent people. The set of beautiful paintings. The set of large whole numbers.

A set may be defined even though it contains no elements: the set of people who have been on Mars contains no elements. The set with no elements can also be defined by {x\x is a whole number less than zero}. Although the set with no elements may be defined in many different ways, there is only one set with no elements in it. We name this set the empty set and seldom use the expression "an empty set." DCFINITION 2-2-2 The set containing no elements is the empty set. The empty set is denoted by the symbol 0. It may also be denoted by a pair of empty braces { }. but the symbol 0 is usually preferred and is easier to write.

2 3 S€T ftelATIONS

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Note that the empty set is not designated by {0}; this notation would represent a set containing the empty set as an element, rather than the set with no elements. Similarly, the set {0} contains the element zero and hence is not the empty set. It is correct, therefore, to write 0e{0} and 0G{0}, but 0£0 and O£0.

2-3

S€T R€lflTIONS Sometimes we find that every member of one set is also a member of some other set. If such a relation exists, the one set is said to be a subset of the other set. For example, the set of all cows in the world is a subset of the set of all mammals in the world.

DEFINITION 2-3-1

Set A is a subset of set B (denoted by AC B) if and only if every element of set A is also an element of set B.

A EXAMPLE 1

If A = {a, b) and B - {a, b, c}, then ACB, because every element in A is also an element in B. This relation may also be expressed by writing {a,b}Q{a,b,c}.

A EXAMPLE 2

If C = {2, 7} and D = {2, 3, 4}, then C is not a subset of D, since 7 G C but 1 $lD. We can then write C £ D, which is read, "C is not a subset of D."

It follows from Definition 2.3.1 that every set is a subset of itself. Furthermore, because the empty set contains no elements, it is necessarily true that there are no elements in the empty set that are not members of every other set; hence, the empty set is a subset of every other set, including itself. (Problem 6 of Exercise 2.3 further clarifies this somewhat elusive concept.) Frequently, we wish to refer to part of a set but not to the entire set. In this case, we are dealing with a proper subset of the original set. The set of cows, for example, is not only a subset of the set of mammals but also a proper subset of the set of mammals, because some mammals are not cows.

DEFINITION 2-3-2

Set A is a proper subset of set B (denoted by A C B) if and only if every element of A is an element of B but there is at least one element of B that is not an

S€TS. R615ITIONS. AND FUNCTIONS

2

element of A. (Note that by this definition every subset of any given set B, except B itself, is a proper subset of B.)

A €XAMPl€ 3

If S = {a, b, c}, the subsets of 5 are {a, b, c), {a, b), {a, c}, {b, c}, {a}, {b}, {c}, and 0. By definition, all of these are proper subsets of 5 except {a, b. c}, which is the set 5 itself.

An effective way to illustrate relations among sets is to use diagrams such as those in Figures 2.3.1 and 2.3.2. Diagrams of this type (which were used in Chapter 1 to illustrate various types of valid arguments) are called Venn diagrams in honor of the English logician John Venn, who first used this type of diagram in his Symbolic Logic (1881). In Venn diagrams, elements of a set are represented by the part of a plane inside a simple closed curve. Sometimes, specific elements of a set are indicated by letters or other symbols within the simple closed curve. In Figure 2.3.1, we see the set of cows (C) represented as a proper subset of the set of all mammals (M); also, the set of mammals is represented as a proper subset of the set of all animals (A). It is important to note in the diagram that all of region C is contained in region A/, but not all of region M is in region C since C is a proper subset of M (CCM). Similarly, region M is contained within region A, since MCA. Is CCA? In Figure 2.3.2, we see that DCU, where D = {a, b< c. d) and U = {a, b, c, d, e,f, g}. Note that each element is placed in the region representing the set to which it belongs. Usually, it is necessary to restrict the elements from which a given set may be chosen. The set that contains all elements being considered in a given discussion is called the universe or the universal set for that particular situation. Referring again to Figure 2.3.1, we see that our discussion was restricted to the set of all animals and that the universe is represented by the letter A. In Figure 2.3.2, the universe is U- {a, b, c, d. e, /, g). The universe is usually represented by the letter U, unless some other letter is more appropriate. If the universe were the set of whole numbers, for example, we might use W to represent the universe. For each set A in any universe U, we can identify another related set known as the complement of A.

FIGUR€

2-3-1

FIGUR€ 2-3-2

37

2 3 S€T R61ATIONS

DCFINITION 2-3-3 For any set .4 in universe L . the complement of set A (denoted by A ) is the set of elements in the universal set U that are no! in A. Using set-builder notation, we write A ={x xEU

A GCAMPIC 4

and x£A)

If U= {0. 1. 2. 3. 4} and .4 = {2. 3}. then the complement of A is .4 = {0. 1. 4}. The Venn diagram below illustrates this example. The shaded region represents A. In studying the diagram, also note that the complement of A is A ={2,3}; or A = A. L

^

(

A 1

2 3

/

4

\ ° ▲ GCAMPIC 5

If U = {x |x is a human} and A = {x x is a human over 6 feet tall}. then A = {jc |x is a human not over 6 feet tall}. The shaded region of the following Venn diagram represents A. If a person is exactlv 6 feet tall, would he or she be in set .4 or A1

€X€RCIS€ 2-3 1. Use the roster method to identify in as many ways as possible the set whose elements are 0. 1. and 2. 2. Which of (a) The (b) The (c) The (d) {x\x (e) {x\x

the following sets are well-defined'! set of whole numbers less than 15. set of fair methods of taxation. set of letters of the English alphabet. is a great living actor} is a living ex-President of the United States}

S€TS

R61ATIONS

RND FUNCTIONS

2

3. Use the roster method to define each of the following sets. If a set has many elements, use three dots to indicate the unnamed elements. (a) The set of whole numbers less than 1000. (b) The set of whole numbers greater than 9 and less than 100. (c) (d) (e) (f) (g)

{x|x + 3 = 5} {y\3y-4=U} iy \)'2 ~ 36 = 0 and y is an integer} {x\x-2n and n is a whole number less than 5} {x\x = 2n + 1 and n is a whole number less than 5}

{y\y2 = 0} (h) set-builder 4. Use notation to define each of the following sets. (More than one method is possible.)

5.

6.

7.

8.

9.

(a) {.... ~2, "1, 0. 1. 2. ...} (b) {4. 6. 8. 10. 12. ...} (c) {0. 1. 4. 9. 16. ...} (d) {o. b. c. d. e. f. g. h. i. /} (e) 0 (f) {20.21.22.23} (g) {0.3.6,9....} (h) {1.4.7.10....} State the number of elements in each of the following sets: (a) {7. o, A. 0} (b) {a. X 0} (c) {A.0} (d) {0} (e) 0 (f) {0} Let B = {a. b, c, d). (a) Is even element of 0 in B? (If your answer is no. name an element of 0 that is not in B.) (b) Is every element of B in 0? (c) Is 0 a proper subset of B by Definition 2.3.2? Which of the following are true? (a) 0G0 (b) 0C0 (c) 0C0 (d) OE0 (e) 0 = {O} (f) 0 = {0) (g) 0 = 0 (h) 0e{O} (i) 0C{O} (j) 0E{0} Given A = {0. 1. 2}. which of the following are true? (a) AC A (b) AC A (c) {0}CA (d) OCA (e) 0CA (f) A = {2. 0,1} (g) 0EA (h) 2CA (i) {2}CA (j) OCA Let A = {0, 1,2}, B = {0,1, 2. 3.4}. and C = {0,1,2, 3, 4. 5. 6. 7): (a) What is A if B is the universe? (b) What is A if C is the universe? (c) What is A if the set of whole numbers is the universe?

(d) What is A if B is the universe"1 If C is the universe1 If W is the universe? 10. Let the universe be the set of all rabbits. (a) What is the complement of the set of all female rabbits?

2 3 S€T RCIATIONS

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(b) What is the complement of the set of all male rabbits? (c) If A is the complement of B, is B necessarily the complement of A? 11. It is given that U is the universe and that ACB and BCU. Illustrate this with a Venn diagram; shade A with horizontal parallel lines and B with vertical parallel lines. 12. (a) If A C 5, what is the least number of elements that could be in Bl Explain, (b) U AC B and BCC, what is the least number of elements that could be in C? Explain. 13. (a) What relation exists between A and C if A C B and BCC? (b) What is the relation between A and C if A C B and B C C? (c) What isthe relation between A and B if A C B and BCA? '14. (a) If A = B, what is Al (b) If U is the universe, what is 0? (c) If U is the universe, what is Ul '15. Given S = {2, {3}, {4, 5}, 5. 6}, which of the following are true? (a) 2ES (b) {2} £5 (c) {2}C5 (d) {3}£S (e) {2, {3}}C5 (f ) The complement of {2. {3}} with respect to set 5 is {{4. 5}. 5. 6}. "16. Let S = {2, 3. {5.7,8}, 8}.

(a) (c) (e) (g) (i) (k)

Is Is Is Is Is Is

{5,7, 8}E5? 5 e 5? {5. 7}es? {3. {5, 7, 8}}C5? {8} C 5? {5,7, 8}C5?

(b) (d) (f) (h) (J)

is 2 Is 8 How Is 8

e s? e 5? many elements are in 5? C 5?

Is 7, 8}}C5? 0C5° Is {{5,

'17. Draw a Venn diagram to illustrate (1) that ACB and BCC. Use this diagram to assist in supplying an appropriate conclusion for each of the following arguments. (Note how the arguments relate to types of valid arguments discussed in Chapter 1: the law of detachment, the law of the contrapositive, and the chain rule.) (a) Premises:

(1)

(2) Conclusion: (b) Premises:

xCA ?

(1)

If x CA. then ,vE B.

(2)

x£/*)}. How many elements are in A x B?

(d) How many elements are in 5 x .4° (e) How manv elements in A x B are also in fix .4° Let .4 ={2. 3, 4} and B = 0. (a) How many elements are in set A? (b) How many elements are in set B? (c) Find {(x. \)\(xEA) and ( v E B)}. (d) Find {(x. y)|(x6B) and (v64)}. (e) How many elements are in A x B? (f) How many elements are in B x A? (g) How many elements are in 0x0? If A = {lt 2, 3}, 6ndAxA={(xty) (x EA) and (y 6/1)}. Let /I = {3. 4}. B = {4. 5. 6}. and C = {5. 6. 7. 8}. (a) Find .4 x(BHC). (b) Find (A x B) n C. (O Find (.4 P B) x C. (d) Find.4n(BxC). Given .4 = {/>. c}. B = {2}. and C = {3. 4. 5}. show by the roster method that A x (B U C) = (/I x fl) u L4 x C). Let A ={1, 2, 3} and £ = {4. 6. 8. 9}. (a) How many elements are in A x B? (b) How many relations are there from .4 to B. excluding the empty set as a relation? (Hint: A set of n elements has 2" subsets: see Problem 18 of Exercise 2.3.) (c) Identify bv the roster method the relation B, from .4 to B. if

fl,={tv. v) v = 2.v}. (d) Identify by the roster method the relation A x B and y = Ix + 2}. 9. Which of the following relations are functions? B,={(2. 4). (3.4). (5,9)} B: = {(1.2). (3.4). (5.6). (7.8)} B, = {(3. 7). (4.9). (4.10)} /?4 = {(1. 70). (2. 70). (3. 70). (4. 70). (5, 70)} ft5 = {(3,5), (3.6). (3.7). (3,8), (3.9)}

B: = {(.v.y) (.v. y)6

2 7 R61ATIONS

AND FUNCTIONS

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10. The velocity of a falling object expressed in feet per second is v = 32t. Using {0, 1, 2, 3} as the domain: (a) Express the relation of the velocity v of a falling object to the time t it has fallen by R = {(/, v)\v = 32?}. (b) Is the velocity of a falling object a function of time; that is, is R a function? (c) (d) (e) 11. Let (a)

What is the velocity when t = 0? What is the velocity when t = 3? What is the range of /?? C = {3, 11,45} and D = {1, 4, 5, 17}. Identify by the roster method R - {(x, y) \(x, y) G (C x D) and x is less than y).

(b) Is the relation R from CtoDa 12.

function?

If f(n) = k3 + 2, find: (a) /(0) (b) /(2)

(c) /(4) (d) /(5) 13. If the domain of f(x) = 3x2 + 2 is {2, 3, 7}, what is the range of the function? 14. The volume of a cube is a function of its edge e and is v =/(e) = e3. (a) What is the volume of a cube with an edge that measures 2 meters? (b) Concrete is often measured and sold by the cubic yard. How many cubic feet of concrete are in a cubic yard? (c) If you must order a whole number of cubic yards of concrete, how many cubic yards should be ordered for a job requiring 100 cubic feet? 15. Mike was a heavy baby, weighing 10 pounds at birth. His parents recorded his weight to the nearest 5 pounds every 3 years until he went away to college at the age of 18. The results are shown on the accompanying graph. 150

100

50

20

10 Age

(a) (b) (c)

15

Express Mike's weight as a function of his age with a set of ordered pairs of the form (a, w), where a is his age and w is his corresponding weight. What is the domain of the function? What is the range of the function?

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S€TS. R€lATIONS

AND FUNCTIONS

2

16. Mike's height (measured in inches) as a function of his age (in years) is expressed by the set {(0. 20), (3. 35). (6, 40). (9. 50). (12. 60). (15. 65), (18, 70)}. Make a graph similar to the one in Problem 15 showing Mike's height as a function of his age. *17. The daily fixed costs of the Clear-Tone Radio Company are $2000. In addition, it costs $6 for each radio produced. (a) Use an equation to express the cost C of producing x radios per day as a function of x. (b) What is the total cost of producing 100 radios in a day? What is the cost per radio? (c) What is the total cost if 1000 radios are produced in a day? What is the cost per radio? (d) If the Clear-Tone Radio Company can sell its radios for $16 each, can it stay in business producing 100 radios per day? 1000 radios per day? (e) How many radios must be sold in a day to "break even" (that is. to have costs equal sales)? 18. If set A has 6 elements and set B has 4 elements: (a) How many ordered pairs are there in A x 5? (b) How many possible relations could be identified from set A to set Bl (Exclude the empty set as a relation.) (c) How many relations are therefrom set B to set A? (Exclude the empty set as a relation.)

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PROP€RTI€S OF RCIRTIONS As mentioned previously, relation on A . If a relation A and the relation RCA Relations on a set interested. In particular,

a relation from any given set A to set A is said to be a R is on A , then both its domain and its range are in set x.A. A often have various properties in which we are we will discuss the reflexive, symmetric, and transitive

properties. DEFINITION 2-81 A relation R defined on a set A is reflexive if and only if for every aGA. a is related to a. Thus, for aEA. (a. a)GR. Note that this definition states that every element in A must be related to itself if the reflexive property is to hold for a given relation R.

▲ CXAMPIC 1

Consider a set of people A and the relation "is the same height as" defined on A. Certainly, each person is the same height as

2 8 PftOP€flTI€S Of fl€lATIONS

59

himself or herself. Then, by definition, '"is the same height as" is a reflexive relation, and "a is related to a" holds for every a E.A. A €XflMPL€ 2

Consider the set B = {1. 2. 3. 4} and the relation "is less than" defined on B. It is not true that even, element is less than itself. (As a matter of fact, no element is less than itself.) Thus, the relation "is less than" is not reflexive.

DEFINITION 2-8-2 A relation R defined on a set A is s\ mmetric if and only if for every a and bin A, whenever a is related to b. then b is related to a. Thus, if {a. b)ER. then (b.a)ER. Note that this definition does not require that any given element a be related to b: however, if a is related to b. then b must be related to a for the relation to be svmmetric.

A €XRMPl€ 3

Consider a set of people A and the relation "is the same height as" defined on .4 . It is true that if a is the same height as b. then b is the same height as a. Because "a is related to b" implies "b is related to a." the relation "is the same height as" is symmetric.

A €XAMPl€ 4

Consider the set B = {1, 2, 3, 4} and the relation "is less than" defined on B. Then "a is less than b" does not imply "b is less than a." Because "a is related to b" does not imply "b is related to a." the relation "is less than" is not symmetric.

DEFINITION 2-8-3 A relation R defined on a set A is transitive if and only if for even a. b. and c in .4. whenever a is related to b and b is related to c. then a is related to c. Thus, if (a. b) n(ADB) = ?

11. Given A = {a. b, c. d} and B = {c. d. e). complete the following: (a) n(A) = ? (b) n(B) = ? (c) n(AUB) = ? (d) n{ADB) = ? (e) «({c,