CLEP College Mathematics

THE BEST TEST PREPARATION FOR THE

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College

Mathematics

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CONTENTS Chapter 1 Passing the CLEP College Mathematics Exam About This Book and TESTware® About the Exam How to Use This Book and TESTware® Format and Content of the Exam About Our Course Review Scoring Your Practice Tests Studying for the CLEP Test-Taking Tips The Day of the Exam

1

3

3

6

6

7

7

8

8

10

Chapter 2 Sets Sets Subsets Union and Intersection of Sets Laws of Set Operations Cartesian Product Drill Questions

11

13

15

17

18

20

22

Chapter 3 The Real Number System Properties of Real Numbers Components of Real Numbers Fractions Odd and Even Numbers Factors and Divisibility Numbers Absolute Value Integers Inequalities Drill Questions

27

vi

29

31

31

32

.32

33

34

36

41

Chapter 4 Alge Exponents.... Logarithms .. Equations..... Simultaneow Absolute Vall Inequalities .. Complex NUl Quadratic Eq Advanced AI. Drill Questio

Chapter 5 FUM Elementary I

Translations.

Drill Questio

Chapter 6 Geol Triangles . The Pythagol Quadrilateral Similar Po1YJ Circles .. Formulas for Drill Questi(J

Chapter 7 Prot The Fundam Permutation! Combinatior Probability ..

Probability " Measures of Measures of Data Ana1ys Analyzing P Drill Questic

Contents

Exam 1

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.......................... 17

.......................... 18

, 20

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........................ 27

.........................29

.........................31

.........................31

.........................32

.........................32

.........................33

.........................34

.........................36

.........................41

Chapter 4 Algebra Topics Exponents Logarithms Equations Simultaneous Linear Equations Absolute Value Equations Inequalities Complex Numbers Quadratic Equations Advanced Algebraic Theorems Drill Questions

45

47

50

52

57

64

65

67

70

76

77

Chapter 5 Functions and Their Graphs Elementary Function : Translations, Reflections, and Symmetry of Functions Drill Questions

83

85

90

93

Chapter 6 Geometry Topics Triangles The Pythagorean Theorem Quadrilaterals Similar Polygons Circles Formulas for Area and Perimeter Drill Questions Chapter 7 Probability and Statistics The Fundamental Counting Principle Permutations Combinations Probability Probability Word Problems Measures of Central Tendency Measures ofVariability Data Analysis Analyzing Patterns in Scatterplots Drill Questions

'"

99

10 1

l 07

110

116

118

124

126

131

133

134

13 5

136

142

144

149

151

156

160

vii

CLEP College Mathematics

Chapter 8 Logic Sentences Statements Basic Principles, Laws, and Theorems Necessary and Sufficient Conditions Deductive Reasoning Truth Tables and Basic Logical Operations Logical Equivalence Sentences, Literals, and Fundamental Conjunctions Drill Questions

165 167 170

Practice Test 1 Answer Key Detailed Explanations ofAnswers

187

Week

203 205

1

Practice Test 2 Answer Key Detailed Explanations of Answers

217 233 235

Answer Sheets

249

Installing REA's TESTware®

262

172 173 l 74 175 178 182 183

The followi CLEP College :N it can be reducel riod into one. & day-to study. l' more time you 51 on the day of the

I Re

yOl

Pn

am

COl

yOl

2&3f'Ca

re\l

bo

~

the an~

- 3." The graph

II

...

...

-5

A half~pen inle

Chapter 3: The Real Number System

Example: {x I x

< 4}. The graph would appear as:

. . . . .--------~---+I--------1.~

o

Ie for a number to be

4

Example: {x

12 < x < 6}. The graph would appear as:

. . ... --------1H---+--------·~

o

2

6

r number must belong such as 3, belong to rately, you will notice rIap between 2 and 5,

Example: {x I x is any real number}. The graph would appear as:

)Dints that satisfy the

It x

o

A closed interval includes two endpoints.

Example: {x

> - 3." The graph

I - 5 -

3}. The graph would appear as:

+

I

III

o

•

3

(A) ~ 2 (B) 3c ­ 1

Example:

{x Ix

1. If c is any odd ill integer?

0 (B) Ix 1= 0 (C) Ix I = -x (D) I x 1< 0

42

Chapter 3: The Real Number System

m.OOr between 90 and

7. The number 0.4 is equivalent to which fraction?

(A)

~

(B)

~

(C)

~

5

7

9

(D)~ 11

l graph

on the number

.true for any x?

8. Not including the number 16, how many factors of 16 are there?

(A) 7 (B) 6

(C) 5 (D) 4

9. n 2 is a proper fraction in reduced form. Which one ofthe following must be true?

(A) n is a proper fraction. (B) n is an improper fraction. (C) n is a positive number. (D) n is a negative number.

10. Which one ofthe following has no solution for x?

(A) Ix

I> 0 (B) Ix 1= 0 (C) Ix I = -x (D) Ix 1< 0 43

(0)

'01

(a)

'~

(y) '£

'8

'v

(y) '6 (0)

(0)

(a)

'9

(0)

(0) 'L (a)

'z 'I

Chapter 4

Algebra Topics This chapter discusses some interesting algebra topics that you should know, such as exponents, logarithms, complex numbers, and inequalities. In addition, it presents applications from algebra so you can see the usefulness of this subject.

EXPONENTS

In the expression 32 ,3 is the base and 2 is the exponent. This means that 3 appears 2 times when multiplied by itself(3 X 3), and the product is 9. An exponent can be either positive or negative. A negative exponent implies a fraction, such that if n is a negative integer a

So 2 ,

-4

I 24

-n

I

= - a=tD an'

I 16

=-=­

We see that a negative exponent yields a fraction that is the reciprocal of the original base and exponent, with the exponent now positive instead of negative. Essentially, any quantity raised to a negative exponent can be

47

CLEP College Mathematics

"flipped" to the other side of the fraction bar and its exponent changed to a positive exponent. For example, 3

I

1. Here the exponent

1

_y

4- = 43 = 64

x

An exponent that is 0 gives a result of 1, assuming that the base itself is not equal to O. aO

= 1, a

~

=/= 0

(­

-3 ali

1

2. In this case, the ex

An exponent can also be a fraction. If m and n are positive integers, m

=­

nr-;;

= '\lam

3. 4­

The numerator remains the exponent of a, but the denominator tells what root to take. For example,

1

m' -3

=

--1­

1

4. -16 -2 --

-

=_

_1_1 = 162

4~=~=~=8 3~=~=h1=9

General Laws Cl

If a fractional exponent was negative, the operation involves the recip­ rocal as well as the roots. For example,

27-~ = _1_ = _1_ = _1_ = 27~

~272

~729

! 9

tfJdI

(c/J

(/J

Simplify the following expressions:

a'l

1. -3- 2 2. (-3)-2

-3

3.­ 1 41

4. -16-2

48

(ab)

Chapter 4: Algebra Topics

q>Onent changed to a 1. Here the exponent applies only to 3. Since xIIg

y

=

~,

1 so -3- 2 = _(3)-2 = - (312 ) = --

9

that the base itself 2. In this case, the exponent applies to the negative base. Thus,

(-3)-2 = _1_ = 1 (-3)2 (-3)(-3) c positive integers,

1

9

-3 -3 -3 -3 -3 4 3 -= = = = -x-=-12 1 . 41 ~ ! 1 1

4 41 4

(!)

the denominator tells

I

4. -16-2

1 =-= 16!

1

1

v%

4

- - - =-­

General Laws of Exponents

00 involves the recip­

aPaq = aP + q, bases must be the same.

4243 = 42 + 3 = 45 = 1,024 1

9

(aP)q

= aPq (2 3)2 = 26 = 64

aP

- q = aP- q , bases must be the same, a i= 0 a

(ab)P

= aPb P

(3

X

2)2 = 32 X 22 = (9)(4) = 36

49

CLEP College Mathematics

aP (ba)p =bP,b*,O

Find the value of

4)2 _ 42 _ 16 ---(5 52 25 -

logs 25 = x is e 3 + y is a conditional inequality for the set of real numbers, since it is true for any replacement less than 0 and false for all others, or y < 0 is the solution set.

The sentence x + 5 > x + 2 is an absolute inequality because the expression on the left is greater than the expression on the right.

e values, proceed as false when its

65

CLEP College Mathematics

The sentence x + 10 < x + 5 is inconsistent because the expression on the left side is always greater than the expression on the right side.

•

~

The sentence 5y < 2y + Y is inconsistent for the set of non-negative real numbers. For any y greater than 0, the sentence is always false. Two inequalities are said to have the same sense if their signs of inequality point in the same direction. The sense of an inequality remains the same ifboth sides are multiplied or divided by the same positive real number.

Solve the inequali1

Example:

For the inequality 4 > 3, if we multiply both sides by 5, we will obtain:

4X5>3x5

Add - 5 to both si Additive inverse p

Additive identity I Combine terms:

20> 15 Multiply both side The sense of the inequality does not change. Ifeach side ofan inequality is multiplied or divided by the same negative real number, however, the sense of an inequality becomes opposite.

same as dividing 1

The solution set iJ Example:

X= {xlx>2}

For the inequality 4 > 3, if we multiply both sides by - 5, we would obtain:

(that is, all x, sud

4 X (- 5) < 3 X (- 5)

COMPLEX NUMB

-20 < -15

As indicated abo' Ius mathematics topic: are not enough to expl were developed.

The sense of the inequality becomes opposite. If a > b and a, b, and n are positive real numbers, then an> bn and a- n < b- n

> y and q > p, then x + q > Y + p. Ifx > y > 0 and q > P > 0, thenxq > yp. If x

66

Chapter 4: Algebra Topics

lOSe the

expression on lie right side. .e set of non-negative ,always false.

_

if their signs of

Ib. sides are multiplied

Solve the inequality 2x

+5>

9.

Additive identity property:

+ 5 > 9. 2x + 5 + (- 5) > 9 + (- 5) 2x + 0 >: 9 + (- 5) 2x> 9 + (-5)

Combine terms:

2x >4

2x

l

sides by 5, we will

d by the same negative JDCS opposite.

ides by -5, we would

s,then

Add - 5 to both sides: Additive inverse property:

Multiply both sides by! (this is the 2 same as dividing both sides by 2):

I I -(2x) > 2 2 x>2

X 4

The solution set is

X= {x

Ix > 2}

(that is, all x, such that x is greater than 2).

COMPLEX NUMBERS As indicated above, real numbers provide the basis for most precalcu­ lus mathematics topics. However, on occasion, real numbers by themselves are not enough to explain what is happening. As a result, complex numbers were developed.

67

CLEP College Mathematics

(3

+

Returning momentarily to real numbers, the square of a real number cannot be negative. More specifically, the square of a positive real number is positive, the square of a negative real number is positive, and the square of 0 is O.

i is defined to be a number with a property that F = - 1. Obviously, i is not a real number. C is then used to represent the set of all complex numbers:

Division of Comil

Division of two ce

cial procedure that invo C

= {a + bi I a and b are real numbers}.

Addition, Subtraction, and Multiplication of Complex Numbers Here are the definitions of addition, subtraction, and multiplication of complex numbers. Suppose x thati2 = - 1):

+ yi and z + wi are complex numbers. Then (remembering

Simplify (3

(x

+ yi) + (z + wi) = (x + z) + (y + w)i

(x

+ yi)

- (z

+ wi) =

(x

+ yi)

X (z

+ wi) = (xz

+ i)(2 + i).

(x - z)

+ (y -

- lry)

gate of a

+ bi is denok

Also,

The usual proced1 jugate as shown beloVll same quantity leaves tI

w)i

+ (xw + yz)i

If a is a real nWII Hence, every real nun

All the propertie complex numbers, so

68

Chapter 4: Algebra Topics

(3

+ i)(2 + i)

are of a real number l positive real number 1Sitiye, and the square l ;2

= -

1. Obviously, be set of all complex

= 3(2

+ i) + i(2 + i)

= 6

+ 3i + 2i + P

= 6

+ (3 + 2)i + (-1)

= 5

+ 5i

Division of Complex Numbers Division of two complex numbers is usually accomplished with a spe­ cial procedure that involves the conjugate of a complex number. The conju­

crs}.

gate of a

+ bi is denoted by a + bi and defined by a 1- bi = a - bi.

Also,

(a

+ bi)(a - bi)

= a2

+ b2

, and multiplication of

L

Then (remembering

The usual procedure for division is to multiply and divide by the con­ jugate as shown below. Remember that multiplication and division by the same quantity leaves the original expression unchanged.

x

+ yi

x

--= z + wi z

.. w)i

- w)i (~+

+ yi z - wi x-­ + wi z - wi

(xz

yz)i

+ yw) + (-xw + yz)i

=--'----;:----;:---'-­ z2 +w2

=

xz+yw z2 + w2

+

-xw+yz. l z2 + w2

If a is a real number, then a can be expressed in the form a = a Hence, every real number is a complex number and R ~ C.

+ Oi.

All the properties of real numbers described in Chapter 2 carry over to complex numbers, so those properties will not be stated again.

69

CLEP College Mathematics

QUADRATIC EQUATIONS

We need two nUll would be + 3 and +4, (x + 4) = O.

x

There are several methods to solve quadratic equations, some of which are highlighted here. The first two are based on the fact that if the product of two factors is 0, either one or the other of the factors equals O. The equation can be solved by setting each factor equal to O. If you cannot see the factors right away, however, the quadratic formula, which is the last method presented, always works.

=

Therefore, x + 3 : -3,x = -4.

Substitute the vall For x = -3, (-3 x = - 3 is a solution. Likewise, for x =

= O. So x = -4 is a sc

Solution by Factoring We are looking for two binomials that, when multiplied together, give the quadratic trinomial

ar + bx + c = 0 This method works easily if a = 1 and you can find two numbers whose product equals.c and sum equals b. The signs of band c need to be considered:

• If c is positive, the factors are going to both have the same sign, which is b's sign. • If c is negative, the factors are going to have opposite signs, with the larger factor having b's sign.

Example:

Suppose the quae: the sign of b is negatn

We need two num be - 3 and -4, and the

Therefore, x - 3 are x = 3, x = 4.

Once you have the two factors, insert them in the general factor format (x

+ _)(x + _) = O.

To solve the quadratic equation, set each factor equal to 0 to yield the solution set for x.

Substitute the va For x = 3, solution.

(3i ­

Likewise, for x : X

Example:

Solve the quadratic equation 70

r

+ 7x + 12 = O.

= 4 is a solution.

Chapter 4: Algebra Topics

We need two numbers whose product is + 12 and sum is +7. They would be +3 and +4, and the quadratic equation would factor to (x + 3) (x

+ 4) = O.

Therefore, x + 3 x = -3,x = -4.

: equations, some of DO the fact that if the rthe factors equals O. 131 to O. If you cannot nola. which is the last

Substitute the values into the original quadratic equation: For x = -3, (-3)2 + 7(-3) + 12 x = - 3 is a solution.

= dtiplied together, give

= 0 or x + 4 = 0 would yield the solutions, which are

= 0, or9

Likewise, for x = -4, (-4)2 + 7(-4) + 12 o. So x = -4 is a solution.

+ (-21) + 12

= O.

So

= Q, or 16 + (-28) + 12

Example: Suppose the quadratic equation is similar to the previous example, but the sign of b is negative. Solve the quadratic equation x 2 - 7x + 12 = o.

:an find two numbers

of b and c need to be same sign, which is

We need two numbers whose product is + 12 and sum is -7. They would be - 3 and -4, and the quadratic would factor to (x - 3)(x - 4) = O.

posite signs, with the

Therefore, x - 3 = 0 or x - 4 = 0 would yield the solutions, which are x = 3,x = 4.

Ie

:general factor format Substitute the values into the original quadratic equation:

equal to 0 to yield the

For x solution. x

=

= 3, (3)2

Likewise, for x 4 is a solution.

- 7(3) + 12

= 4, (4)2

= 0, or 9

- 7(4) + 12

So x

= 3 is a

- 28 + 12

= O. So

- 21 + 12

= 0, or 16

= O.

71

CLEP College Mathematics

Example: As a final example, solve the quadratic equation Y?

+ 4x - 12 = 0.

For x For x

We need two numbers whose product is -12 and sum is +4. They would be +6 and -2 (note that the larger numeral gets the + sign, the sign of b). The quadratic equation would factor to (x + 6)(x - 2) = 0.

°

Therefore, x + 6 = are x = -6,x = 2.

or x - 2 =

°

Likewise, for x x = 2 is a solution.

+ 4(-6) - 12 = 0, or 36 + (-24) - 12 = 0. So

= 2,

(2i

Example: Solve the quadrat

would yield the solutions, which

Substitute the values into the original quadratic equation: For x = -6, (-6i x = -6 is a solution.

= 4, (4)2 ­ = -4 , (-4'.

This is the diffeR roots are 3x and 6. So 1 or 3x - 6 = 0, and the

+ 4(2) - 12 = 0, or 4 + 8 - 12 = 0. So For x

Sum of Two Squares

For x = -2,9(­

If the quadratic c~nsists of the difference of only two terms of the form aY? - c, and you can recognize them as perfect squares, the factors are sim­ ply the sum and difference of the square roots of the two terms. Note that a is a perfect square, but not necessarily 1, for this method.

Quadratic Forml quad! of quadratic. as shoWn b

Example: Solve the quadratic equation Y?

= 2, 9(2)2 .

-

16

= 0. (til-

This is the difference of two perfect squares, Y? and 16, whose square roots are x and 4. So the factors are (x + 4)(x - 4) = 0, and the solution is x = ±4.

72

Chapter 4: Algebra Topics

-l + 4x ­

12 = O.

For x For x

md sum is +4. They Is the + sign, the sign ~ - 2) = O.

= 4, (4)2 - 16 = 16 - 16 = O. = -4, (-4)2 ­ 16 = 16 - 16 = O.

Example:

Solve the quadratic equation

I the solutions, which

~on:

. (-24) ­ 12 = O. So

.. + 8 -

9~ - 36

= O.

This is the difference of two perfect squares, 9x4 and 36, whose square roots are 3x and 6. So the factors are (3x + 6)(3x - 6) = 0, then 3x + 6 = 0 or 3x - 6 = 0, and the solution is x = ±2.

12 = O. So For x

= 2, 9(2)2 -

For x

= -2, 9( -2)2 ­ 36 = 36 - 36 = O.

36

= 36 -

36

= O.

two terms of the form

~ the factors are sim­ two terms. Note that a

Quadratic Formula

IOd.

and 16, whose square : 0, and the solution is

where (if - 4ac~) iscaUed the 4:1Isc~rIm_nt of thequadra1tic equation.

73

CLEP College Mathematics

• Ifthe discriminant is less than zero (b 2 - 4ac < 0), the roots are complex numbers, since the discriminant appears under a radical and square roots of negatives are imaginary numbers. A real number added to an imagi­ nary number yields a complex number. • Ifthe discriminant is equal to zero (b 2 equal.

-

4ac

= 0), the roots are real and

• If the discriminant is greater than zero (b 2 - 4ac > 0), then the roots are real and unequal. The roots are rational if and only if a and b are rational and (b 2 - 4ac) is a perfect square; otherwise, the roots are irrational.

Since the discrimin unequal.

1. 4xl - l2x

2. 3x

2

3. 5xl

-

+9=0

9

=0

Here, a, b, and c an a

= 5, b = 2, and c

Therefore, b2

-

4ac

= 22 - 4C

Since the discrimin roots are irrational.

+ 3x + 5 =

4. xl

Compute the value ofthe discriminant and then determine the nature of the roots of each of the following four equations:

+ 2x -

3. 5xl

0

Here, a, b, and c aI1 a

= 1, b = 3, and c

Therefore,

7x - 6 = 0

b2

+ 2x - 9 = 0

-

4ac = 32

-

4

Since the discrimin

4. xl + 3x + 5 = 0

t~~ Solve the equatic 2

1. 4x -

l2x

+ 9 = 0,

Here, a, b, and c are integers:

= 4, b = -12, and c = 9.

a

In this equation, l formula.

Therefore, b2

-

4ac

= (-12)2 -

4(4)(9)

=

144 - 144

=0

Since the discriminant is 0, the roots are rational and equal. 2. 3xl - 7x - 6 = 0

Here, a, b, and c are integers: a

= 3, b = -7, and c = -6.

Therefore, b2

74

-

4ac

= (-7)2 -

4(3)(-6)

= 49 + 72 =

121

=

11 2.

J

Chapter 4: Algebra Topics

the roots are complex meal and square roots ~ added to an imagi-

Since the discriminant is a perfect square, the roots are rational and unequal. 3. 5x2

+ 2x -

9 = 0

Here, a, b, and c are integers: I

the roots are real and

• 0), then the roots are if a·and b are rational oots are irrational.

= 5, b = 2, and c =

a

Therefore, b2

IDd equal.

-

4ac

= 22 -

4(5)(-9)

=4+

= 184.

180

Since the discriminant is greater than zero, but not a perfect square, the roots are irrational and unequal. 4. x 2

letennine the nature of

-9

+ 3x + 5 = 0

Here, a, b, and c are integers:

= 1, b = 3, and c = 5

a

Therefore, b2

-

4ac

= 32 -

4(1)(5)

=9-

20

=

-11

Since the discriminant is negative, the roots are complex.

Solve the equation x 2

In this equation, a formula.

-

x

+ 1 = O.

= 1, b = -

x=

1, and c

-( -1) ±

= 1. Substitute into the quadratic

J( _1)2 -

4(1)(1)

2(1) 1 ±.JI=4

=---­ 2

±.J=3

1 =--­ 2

75

CLEP College Mathematics

DRILL QUESTION

1 ± ,,[3i

2

x=

1 + ,,[3i

2

1 - ,,[3i orx=--­ 2

ADVANCED ALGEBRAIC THEOREMS A. Every polynomial equationf(x) = 0 of degree greater than zero has at

least one root either real or complex. This is known as the fundamental theorem of algebra.

B. Every polynomial equation of degree n has exactly n roots.

C. If a polynomial equationf(x) = 0 with real coefficients has a root a + bi, then the conjugate of this complex number a - bi is also a root of f(x) = O. D. If a +../b is a root of polynomial equationf(x) = 0 with rational coef­ ficients, then a - ../b is also a root, where a and b are rational and ../b is irrational.

E. If a rational fraction in lowest terms ~ is a root of the equation a"x" + c an_1XZ- 1 + ... +alx+aO = 0, ao =1= 0, and the aj are integers, then b is a factor of ao and c is a factor of an'

F. Any rational roots of the equation XZ + qlXZ- 1 +q2XZ-2 + ... + qn-IX + qn = 0 must be integers and factors of qn. Note that qI, q2, ..., qn are integers.

76

1. Which one of the j of5x + 9y = 14? 5 (A) y= -x+ 14 9 5 (B) y=--x-l' 9 9 (C) y = - -x + 1, 5 9 (D) y = - -x - 1, 5

2. The solution set of

(A) X = {x Ix

-< ;

(B) X = {x I x

>:

(C) X = {xlx-:

Chapter 4: Algebra Topics

DRILL QUESTIONS

1. Which one ofthe following is an equation of a line parallel to the graph of5x + 9y = 14? 5 (A) y = 9x + 14 r:ater than zero has at m as the fundamental

'II

9 (C) y= -S-x+ 14

roots.

cients has a root a + - bi is also a root of

: 0 with rational coef­ are rational and ,Jb is

9

(D) y= -S-x-14

2. The solution set of the inequality 5 - 7x > -9 is (A) X = {x I x -< 2}

f the equation

Ire

5 (B) y= --x-14 9

an>!' +

integers, then b is a

_~-2 ~ 1hat

+ ... + qn-l x

qI, q2, ... , qn are

(B) X = {x I x > 2}

(C) X = (D) X

{XIX -< ;

}

= {xIX ~;}

77

CLEP College Mathematics

3.

Whatisthevalueof(:-~)

(-3)-2?

6. What is the simplif

(A) lli (A)

-~

(B) -lli

3 1 (B) - ­ 9

(C)

~

(D)

~

(C) i (D) -i

9

7. If x

+ 2 is a factor

3

(A) 22

4. If log 10 P = 0.5 and lOglO (f = 1.2, what is the value of lOglO PQ ?

(B) 16

(A) 1.9

(C) -16

(B) 1.7

(D) -22

(C) 1.1 (D) 0.6

5. What is the solution set for x in the following inequality? 2

(A) 28 - 4x

3

(B) 'f8 - 5x

-6 which appears five ti

: I

Comparing the r of Distributions

It is sometimes 1 the mean, median. ar having to calculate th Step 1: If a bar graph

in the problem. The graph will approximately norma the other. • A graph skewed te The order of the th order). • A graph skewed to The order of the 1 alphabetical order) • A graph that is ap: like Figure 3, and t

146

Chapter 7: Probability and Statistics

SOLUTION

The mean is the value obtained by adding all the measurements and dividing by the number of measurements. m of the set. In fact,

500 + 600

+ 800 + 800 + 900 + 900 + 900 + 900 + 900 + 1,000 + 1,100 11

~rs

are repeated. For 9,300 = 845.45

= 11

1

the set of values.

The median is the value appearing in the middle. We have 11 values, and they are already in order, so here the sixth, 900, is the median. The mode is the value that appears most frequently. That is also 900, which appears five times.

:curs twice whereas

llled bimodal), or in

Comparing the Mean, Median, and Mode in a Variety of Distributions It is sometimes useful to make comparisons about the relative values of

the mean, median, and mode. This section presents steps to do that without having to calculate the exact values of these measures. If a bar graph is not provided, sketch one from the information given in the problem.

Step 1.:

since each value is appears only once, . Of course, in this ency. and the mean

The graph will either be skewed to the left, skewed to the right, or approximately normal. A skewed distribution has one of its tails longer than the other. • A graph skewed to the left will look like Figure 1; its left tail is longer. The order of the three measures is mean < median < mode (alphabetical order).

m. and mode. 1.100

• A graph skewed to the right will look like Figure 2; its right tail is longer. The order of the three measures is mode < median < mean (reverse alphabetical order). • A graph that is approximately normal (also called bell-shaped) will look like Figure 3, and the mean = median = mode.

147

CLEP College Mathematics

PROBLEM

Estimate the 1 how the mean com] - ­ Mode

Mode-­

Figure 1

Figure 2

Mean Median Mode Figure 3

Step 2: Write the word "mode" under the highest column of the bar graph,

because the mode is the most frequent. If the graph is skewed right or left, the positioning ofthe mode establishes the order for the remaining two terms according to the information provided above. If the graph is approximately symmetrical, the value of all three terms are approximately equivalent.

3

2

1

PROBLEM

On a trip to the Everglades, students tested the pH ofthe water at differ­ ent sites. Most ofthe pH tests were at 6. A few read 7, and one read 8. Select the statement that is true about the distribution of the pH test results.

o

1

2

3

A. The mode and the mean are the same.

R The mode is less than the mean.

e. The median is greater than the mean. I

D. The median is less than the mode.

SOLUTION

Sketch a graph. The graph is skewed to the right. The mode is furthest left. Thus, mode

< median < mean. Choices (A), (C), and (D) do not coincide with what has been estab­ lished in terms of relative order. (B) is the only choice that does follow from our conclusions. Thus, (B) is the correct response.

SOLUTION

Graph (A) is sy be about 0, since it ~l with a median of abc of 4-7 than it will Graph (C) is left-ske1 toward the lower Yall few extreme high Yal slightly larger due to

MEASURES OF ,

In addition to IT described with meaSl where the middle of manufacturer of light

148

Chapter 7: Probability and Statistics

~

PROBLEM Estimate the median of each distribution shown below and describe how the mean compares to it.

'.lean

'.~edian

'.lode igure 3

of the bar graph, right or left, ;:>maining two terms )h is approximately ely equivalent. ill

k~wed

C

j

the water at differ­ one read 8. Select t~st results.

st left. Thus, mode

32101234 (A)

o

012345678 (C)

-2

1

2

3

4

5

678

(~

o

2 (0)

4

6

SOLUTION Graph (A) is symmetric with a median at about 0. The mean will also be about 0, since it approximates a normal curve. Graph (B) is right-skewed with a median of about 3. The mean will be pulled more toward the values of 4 -7 than it will be toward 1- 2, so the mean will be greater than 3. Graph (C) is left-skewed with a median of about 5. The mean will be pulled toward the lower values, so it is less than 5. Graph (D) is symmetric with a few extreme high values. The median is about 0, and the mean will be just slightly larger due to the outliers around 6.

at has been estab­ [ does follow from

MEASURES OF VARIABILITY In addition to measures of central tendency, distributions need to be described with measures of variability, or spread. It is not enough to know where the middle of a distribution is, but also how spread out it is. A manufacturer of light bulbs would like small variability in the amount of

149

CLEP College Mathematics

hours the bulbs will likely burn. A track coach who needs to decide which athletes go on to the finals may want larger variability in heat times because it will be easier to decide who are truly the fastest runners.

The range of a data set is simply the difference between the maximum value and the minimum value.

The range is rarely a good choice to represent the data set, especially because it can be affected by outliers. Look, for example, at graph (D) ofthe last problem. Its range is - 3 to + 7, but that doesn't show that almost all of the data lie between - 2 and + 2. For data that are fairly symmetric, the standard deviation and variance are useful measures of variability.

Mar 2.1

Compute the \,

SOLUTION To compute the the mean. In this cia follows: S2

= _I_

n - 1

The variance tells us how much variability exists in a dis­ tribution. It is the "average" of the squared differences between the data values and the mean. The variance is calculated with the formula

1

--12 ­

+ ... ­ 1

= _[Ol

11

.

= 0.08'+

where n is the number of data points, Xi represents each data value, and x is the mean. The standard deviation is the square root of the variance. The formula for the standard deviation is therefore

The standard

j

,)0.084 = 0.290. Ihu

rainfall in Birminaha 1::

from that each month

The standard deviation is used for most applications in statistics. It can be thought of as the typical distance an observation lies from the mean.

DATA ANALYSIS

Data analysis oft.

such as bar graphs. lir more intuitive underst

PROBLEM The average monthly rainfall in inches in Birmingham, England, is shown in the table below.

Bar Graphs

Bar graphs are It

The following bar gr. 150

Chapter 7: Probability and Statistics ~ds to h~at

[1

decide which times because

crs. ~

between

Compute the variance and standard deviation of the monthly rainfall.

SOLUTION

, data set, especially ~. at graph (D) ofthe )\\ that almost all of ~iation and

To compute the variance and standard deviation, we must first compute 7 · data set, x = 26.5 Th e vanance . .IS compute d as = 2.22. the mean. In th IS follows: 12

variance s2

s in a dis­ fferences 3riance is

1 n-1L-­ 1

= - - '" (Xi

- x)2

-[(2.3 - 2.225)2 12 - 1

+ ... + (2.6

+ (1.9 -

2.225)2

+ (2.1

- 2.225)2

- 2.225)2]

1 11 0.084

= -[0.9225] =

mts each ~ation is

The standard deviation is the square root of the vanance, or JO.084 = 0.290. Thus, from this data set, we can say that the monthly mean

"e

rainfall in Birmingham is 2.225 inches and it varies by about 0.29 inches from that each month, sometimes more, sometimes less.

DATA ANALYSIS

s in statistics. It can from the mean.

Data analysis often involves putting numerical values into picture form, such as bar graphs, line graphs, and circle graphs. In this manner, we gain a more intuitive understanding of the given information.

19ham, England, is

Bar Graphs Bar graphs are used to compare amounts of the same measurements.

The following bar graph compares the number of bushels of wheat and

151

l

_C_L_E_P_C_o_"_eg_e_M~a_th_e_m_a_tl_'c_s

~_ _~

corn produced on a farm from 1975 to 1985. The horizontal axis for a bar graph consists of categories (e.g., years, ethnicity, marital status) rather than values, and the widths of the bars are uniform. The emphasis is on the height of the bars. Contrast this with histograms, discussed next.

PROBLEM According to the graph below, in which year was the least number of bushels of wheat produced?

J

~_------------.j Histograms

A histogram i primarily for contil wide spread. The h be of uniform size. the bar denotes the

The histogram on a ship during a r

300..,----------------------­

...

250

+---------r...,--~------.-:::r-----­

200

+E'I_--B_----f~_____E'I____,...",-B______f3______E'I____B-B_­

Example: 10 'c.O

c

"0 C

8

Q)

D. U) '+­

o

6

>,

100

+€::t--___€3r-.g....,.---j:3_-€::t--~,.....,__8'~3__:-I==I----€3,.....,-.g..f:-­

()

c

Q)

I-:

4

::J

cr

~ u..

2

rr1 19751976 1977 1978 1979 1980 1981 198219831984 1985

I

Wheat

0

Pa

Corn

Number of bushels (to the nearest 5 bushels) of wheat and corn produced by Farm RQS, 1975-1985

SOLUTION By inspecting the graph, we find that the shortest bar representing wheat production is the one for 1976. Thus, the least number of bushels of wheat produced in 1975-1985 occurred in 1976.

The intervals h; $500, two spent ben' graph the precise arr The distributior to $620. The data arc observations will be is about $180, but th $620. There are no e

Line Graphs Line graphs ar related subjects. Lin 152

Chapter 7: Probability and Statistics

izontal axis for a bar tal status) rather than lhasis is on the height ~" t.

the least number of

Histograms A histogram is an appropriate display for quantitative data. It is used primarily for continuous data, but may be used for discrete data that have a wide spread. The horizontal axis is broken into intervals that do not have to be of uniform size. Histograms are also good for large data sets. The area of the bar denotes the value, not the height, as in a bar graph. The histogram below shows the amount of money spent by passengers on a ship during a recent cruise to Alaska.

Example: F'"

~

10 'QJ)

c

'6 c

8

OJ

Q.

(f)

'0

6

>,

u

c

OJ

4

:::J

cr ~

I.L..

2

480 500 520 540 560 580 600 620 640 660 '984 1985

Spending

Passenger spending during cruise to Alaska

corn produced by

it bar representing mber of bushels of

The intervals have widths of $20. One person spent between $480 and $500, two spent between $500 and $520, and so on. We cannot tell from the graph the precise amount each individual spent. The distribution has a shape skewed to the left with a peak around $600 to $620. The data are centered at about $590-this is about where half of the observations will be to the left and half to the right. The range of the data is about $180, but the clear majority of passengers spent between $560 and $620. There are no extreme values present or gaps within the data.

Line Graphs Line graphs are very useful in representing data on two different but related subjects. Line graphs are often used to track the changes or shifts 153

---------t----------­

_C_LE_P_C_o_"e_9_e_M_a_t_h_em_at_ic_s

in certain factors. In the next problem, a line graph is used to track the changes in the amount of scholarship money awarded to graduating seniors at a particular high school over the span of several years.

how a family's bu percentages.

PROBLEM PROBLEM

According to the line graph below, by how much did the scholarship money increase between 1987 and 1988?

Using the bue month would plan

$250,000

Ho",

$200,000

$150,000

$100,000

$50,000

Gifts/Crla

81

82 83 84

85 86 87 88 89 90

Amount of scholarship money awarded to graduating seniors, West High, 1981-1990

3

ii

SOLUTION

To find the increase in scholarship money from 1987 to 1988, locate the amounts for 1987 and 1988. In 1987, the amount of scholarship money is halfway between $50,000 and $100,000, or $75,000. In 1988, the amount of scholarship money is $150,000. The increase is thus $150,000 - 75,000 = $75,000.

Pie Charts Circle graphs (or pie charts) are used to show the breakdown of a whole picture. When the circle graph is used to demonstrate this breakdown in terms of percents, the whole figure represents 100% and the parts of the circle graph represent percentages of the total. When added together, these percentages add up to 100%. The circle graph in the next problem shows

154

SOLUTION

To find the am centage allotted to h The family plans to

Stemplots

A stemplot. als, variate data as well. forms a plot much It a class of 32 student:

Chapter 7: Probability and Statistics

is used to track the to graduating seniors rs.

how a family's budget has been divided into different categories by using percentages.

PROBLEM 1

did the scholarship

Using the budget shown below, a family with an income of $3,000 a month would plan to spend what amount on housing? Auto - 15%

r­

19

90

Miscellaneous - 8%

ng seniors,

987 to 1988, locate . scholarship money n 1988, the amount S150,000 - 75,000

:akdown of a whole this breakdown in ld the parts of the ded together, these ~xt problem shows

Family budget

SOLUTION To find the amount spent on housing, locate on the pie chart the per­ centage allotted to housing, or 30%. Then calculate 30% of $3,000 = $900. The family plans to spend $900 on housing.

Stemplots A stemplot, also called stem-and-leaf plot, can be used to display uni­ variate data as well. It is good for small sets of data (about 50 or less) and forms a plot much like a histogram. This stemplot represents test scores to:­ a class of 32 students.

-:..: ~

CLEP College Mathematics

3

3

4

the response variable been observed.

5 6

379

7

22 57

8

126888999

900013345567 10 Key: 6

0 0 0 0 3 represents a score of 63 Test scores

The values on the left of the vertical bar are called the stems; those on the right are called leaves. Stems and leaves need not be tens and ones-they may be hundreds and tens, ones and tenths, and so on. A good stemplot always includes a key for the reader so that the values may be interpreted correctly.

PROBLEM

Describe the distribution of test scores for students in the class using the stemplot above.

SOLUTION

The distribution of the test scores is skewed toward lower values (to the left). It is centered at about 89 with a range of 67. There is an extreme low value at 33, which appears to be an outlier. Without it, the range is only 37, about half as much. The test scores have a mean of appropriately 85.4, a median of 89, and a mode of 100.

PROBLEM

Data collected j grams of fat per sery: which are the explar tionship between fat

SOLUTION

The explanator: calories. The number calories in the snack.

Scatterplots can examined univariate I tion's shape, center. : we focus on its shap~ unusual features. Bel history of the Nationa 1 of the 30 players. games, is noted.

'0

~

o (/; ~

'6 Q.

ANALYZING PATTERNS IN SCATTERPLOTS consist of two variables. Typically, we are looking for an association between these two variables. The variables may be categori­ calor quantitative; in this section, we focus on quantitative bivariate data. Scatterplots are used to visualize quantitative bivariate data.

40000· 35000 . 30000 . 25000 . 20000· 15000 • 10000 • 5000 ­ 4

Bivariate data

The two variables under study are referred to as the explanatory variable (x) and the response variable (y). The explanatory variable explains or predicts

156

The shape of a pI The direction of a scat the explanatory varial of the data. The streni scatterplot are.

d rhe stems; those on . rens and ones-they :m. A. good stemplot s may be interpreted

!rs in rhe class using

lTd lower values (to There is an extreme it. rhe range is only .ppropriately 85.4, a

we are looking for :5 may be categori­ ni\'e bivariate data. lara. ~xplanatory variable

e.iplains or predicts

Chapter 7: Probability and Statistics

the response variable. The response variable measures the outcomes that have been observed.

PROBLEM Data collected from the labels on snack foods included the number of grams of fat per serving and the total number ofcalories in the food. Identify which are the explanatory and response variables when looking for a rela­ tionship between fat grams and calories.

SOLUTION The explanatory variable is grams of fat and the response variable is calories. The number of grams of fat would be a predictor of the number of calories in the snack. Scatterplots can tell us if and how two variables are related. When we examined univariate data in the preceding sections, we described a distribu­ tion's shape, center, spread, and outliers/unusual features. In a scatterplot, we focus on its shape, direction, and strength, and we look for outliers and unusual features. Below is a scatterplot of the top 30 leading scorers in the history ofthe National Basketball Association (NBA). Each point represents 1 of the 30 players. Michael Jordan, who scored 32,292 points in 1,072 games, is noted. Michael Jordan

40000 1:l 35000 ~ 30000 o (A 25000 ~ 20000

,

§ 15000 ~

c.. 10000 5000

"'" .'

.:•.•. .. '" i i i

I

400

600

800

1000

1200

I

I

1400

1600

Games Played NBA Top 30 Scorers

The shape of a plot is usually classified as linear or nonlinear (curved). The direction of a scatterplot tells what happens to the response variables as the explanatory variable increases. This is the slope of the general pattern of the data. The strength describes how tight or spread out the points of a scatterplot are. 157

1

------=---------------------.;t CLEP College Mathematics

,I

,I

The three scatterplots below show comparisons of various directions of the data. The top two have a clear linear trend, whereas the third scatter­ plot shows a random type of distribution with no clear association among points.

The scatterpll the general pattern the outlier, which r

.

' • .

CLEP College Mathematics

DRILL QUESTIONS

1. What is the value of the median in the following sample? {19, 15,21,24, 11 }

3. Mrs. Smith tea tered to her c1< between 80 an 79. Which one dents' exam sc (A) The media

(A) 18

(B) The mean

(B) 19

(C) The media

(C) 20

(D) The medial

(D) 21

2. The sample standard deviation, s, of a group of data is given by the

=

L?- 1 (Xi -

X)2

.

_

4. A jar contains ~ that a randomh

X

(A) _1 20

represents the mean, and n represents the number of data points. What is

(B) _1 10

the sample standard deviation for the following data (rounded off to the

(C) ~ 5

formula S

I -

n-1

,

where

Xi

represents each data pomt,

nearest hundredth)? {12, 15,20,21} (D)

(A) 3.67

~

4

(B) 3.81

(C) 4.24 (D) 8.25

5. From four textl physics, a resea textbook. How r (A) 12 (B) 18

(C) 24 (D) 36

160

Chapter 7: Probability and Statistics

.. - - - - - - - - - - - - - - - - - - - - - - - ­

~

3. Mrs. Smith teaches a class of 150 students. On a recent exam adminis­ tered to her class, 100 students scored 90 or better, 40 students scored between 80 and 89, and the remaining students scored between 70 and 79. Which one of the following statements is correct concerning the stu­ dents' exam scores?

pIc? {19, 15,21,24,

(A) The median equals the mean. (B) The mean is less than the mode.

(C) The median is less than the mean. (D) The median is greater than the mode.

4. A jar contains 20 balls, numbered 1 through 20. What is the probability that a randomly chosen ball has a number on it that is divisible by 4?

ata is given by the ~

each data point, i

(A) _1

20

iata points. What is

(B) _1

10 (rounded off to the

(C) ~ 5 1 (D) ­ 4 5. From four textbooks of college math and three textbooks of college physics, a researcher will choose two math textbooks and one physics textbook. How many different groupings of three textbooks are there? (A) 12 (B) 18

i

!~

I

~M

(D) 36

11

ii fI

I~

i~

161

1

j ".!'

'I

!

CLEP College Mathematics

6. Four out of five dentists surveyed recommended professional teeth cleaning twice a year. Of the dentists who made this recommendation, 10% said they are underpaid. If a dentist from this survey is randomly selected, what is the probability that this selected dentist has recom­ mended professional teeth cleaning twice a year and also said that he or she is underpaid?

(A)~ (B)

!

8. In a class of 30 majors, and 40: female students (A) 16

(B) 14

25

(C) 12

4

(D) 10

15

(C) ~ 5 (D)

1

2­

10

7. An "unusual" number cube has five faces, numbered 1 through 5. This number cube will be rolled twice. Assuming that each of the five num­ bers has the same likelihood of appearing, what is the probability that at least one of the following events will occur? Event 1: The number 3 will appear on the first roll.

9. The mean weigl When two peopl is 166 pounds. \ who leave the ro (A) 136

(B) 145 (C) 154 (D) 163

Event 2: An even number will appear on the second roll.

(A)

~

5

(B) ~ 5

10. At the MNP cor number contains sive. The left-me how many differ

(C) ~ 25

(A) 32,768

(D)~

(B) 8192

25

(C) 6720 (D) 4096

162

Chapter 7: Probability and Statistics

d professional teeth his recommendation, s suryey is randomly j dentist has recom­ d also said that he or

8. In a class of 30 students, 24 are female. There are a total of 12 history majors, and 4 of these history majors are male. How many non-history female students are there? (A) 16 (B) 14

(C) 12 (D) 10

9. The mean weight of a group of ten people in a room is 160 pounds. When two people leave, the mean weight of the remaining eight people is 166 pounds. What is the mean weight, in pounds, of the two people who leave the room? (A) 136

'd 1 through 5. This ch of the five num­ e probability that at

(B) 145 (C) 154 (D) 163

roll. 10. At the MNP company, each person gets an identification number. The number contains five digits, chosen from the digits 1 through 8, inclu­ sive. The left-most digit must be 2. If repetition of any digit is allowed, how many different identification numbers are possible? (A) 32,768 (B) 8192

(C) 6720 (D) 4096

i Ii

i

163

CLEP College Mathematics

Answers to Drill Questions

1.

(B)

6. (A)

2.

(C)

7. (C)

3. (B)

8. (A)

4. (D)

9. (A)

5. (B)

10. (D)

Chapter 8

Logic The topic of logic encompasses a wealth of subjects related to the principles of reasoning. Here, we will be concerned with logic on an ele­ mentary level, and you will recognize the principles introduced here because not only have you been using them in your everyday life, you also have seen them in one form or another in this book. The chapter is designed to famil­ iarize you with the terminology you might be expected to know as well as thought processes that you can use in everyday life. Sentential calculus is the "calculus of sentences," a field in which the truth or falseness of assertions is examined by using algebraic tools. We will approach logic from a "true" or "false" perspective here. This chapter con­ tains many examples of sentences to illustrate the terms that are defined.

SENTENCES A sentence is any expression that can be labeled either true or false.

Examples:

Expressions to which the terms "true" or "false" can be assigned include the following: 1. "It is raining where I am standing." 2. "My name is George."

3. "1 + 2

=

3"

767

CLEP College Mathematics

i

--------------------------f---------­ '!

Examples:

Logical Truth

Expressions to which the terms "true" or "false" cannot be assigned include the following:

A sentence is false; that is, the del

1. "I will probably be healthier if I exercise." 2. "It will rain on this day, one year from now." 3. "What I am saying at this instant is a lie." Sentences can be combined to form new sentences using the connec­ tives AND, OR, NOT, and IF-THEN.

Examples:

Example:

Either Mars is ,

Logical Falsity

A sentence is II true; that is, the sem

The sentences 1. "John is tired." 2. "Mary is cooking."

Example:

Mars is a plane!

can be combined to form 1. "John is tired AND Mary is cooking."

Logical Indetermin

2. "John is tired OR Mary is cooking."

A sentence is II neither logically true

3. "John is NOT tired." 4. "IF John is tired, THEN Mary is cooking."

Example:

Logical Properties of Sentences

Einstein was a r

Consistency

A sentence is consistent if and only if it is possible that it is true. A sentence is inconsistent if and only if it is not consistent; that is, if and only if it is impossible that it is true.

Example:

"At least one odd number is not odd" is an inconsistent sentence.

168

Logical Equivalent

Two sentences a one ofthe sentences t only if it is impossibl

Example:

"Chicago is in I equivalent to "Pittsbu

Chapter 8: Logic 1100

Logical Truth

cannot be assigned

A sentence is logically true if and only if it is impossible for it to be false; that is, the denial of the sentence is inconsistent.

Example:

Either Mars is a planet or Mars is not a planet. cos using the connec­ Logical Falsity

A sentence is logically false if and only if it is impossible for it to be true; that is, the sentence is inconsistent.

Example:

Mars is a planet and Mars is not a planet.

Logical Indeterminacy (Contingency)

A sentence is logically indeterminate (contingent) if and only if it is neither logically true nor logically false.

Example:

Einstein was a physicist and Pauling was a chemist.

'e that it is true. A that is, if and only

~nt

sentence.

Logical Equivalent of Sentences

Two sentences are logically equivalent if and only if it is impossible for one ofthe sentences to be true while the other sentence is false; that is, if and only ifit is impossible for the two sentences to have different truth values.

Example:

"Chicago is in Illinois and Pittsburgh is in Pennsylvania" is logic':":'::­ equivalent to "Pittsburgh is in Pennsylvania and Chicago is in Illinois'-­

1 iii!

CLEP College Mathematics

•

Inverse

STATEMENTS

The inverse of ( A statement is a sentence that is either true or false. but not both.

The following terms and their definitions should become familiar to you. Their logic is probably familiar, even though you haven't given it a label before.

Biconditional

The statement I called a biconditiona

Validity Conjunction

If a and b are statements, then a statement of the form "a and b" is called the conjunction of a and b, denoted by a 1\ b.

An argument i:,; sions must also be tr

Intuition Disjunction

Intuition

The disjunction of two statements a and b is shown by the compound statement "a or b," denoted by a V b.

The negation of a statement q is the statement "not q," denoted by ~q.

PROBLEM

1. If a person is steal

2. If a line is perpenc lar bisector of the

Implication

The compound statement "if a, then b," denoted by a - b, is called a conditional statement or an implication. "If a" is called the hypothesis or premise of the implication, "then b" is called the conclusion of the implica­ tion. Further, statement a is called the antecedent of the implication, and statement b is called the consequent of the implication. Converse

The converse of a - b is b - a. Contrapositive

170

~

Write the imel whether the inverse i

Negation

The contrapositive of a - b is

is the

~b

-

~a.

3. Dead men tell no .

SOLUTION

The inverse of both the hypothesis a

1. The hypothesis of is "he is breaking· not stealing." The. the law." The inverse is fa stealing. Clearly. a the law.

Chapter 8: Logic

Inverse

The inverse of a

!

-+

b is

~a -+ ~b.

or false, but

Biconditional

The statement of the form "p if and only if q," denoted by p called a biconditional statement.

lId become familiar to hayen't given it a label

+-+

q, is

Validity

An argument is valid if the truth of the premises means that the conclu­ sions must also be true.

:he form "a and b" is

Intuition Intuition

is the process of making generalizations on insight.

wn by the compound PROBLEM

Write the inverse for each of the following statements. Determine whether the inverse is true or false. "not q," denoted by

1. If a person is stealing, he is breaking the law. 2. If a line is perpendicular to a segment at its midpoint, it is the perpendicu­ lar bisector of the segment. 3. Dead men tell no tales.

Jy a -+ b, is called a ~d the hypothesis or !Sion of the implica­ :he implication, and

SOLUTION

The inverse of a given conditional statement is formed by negating both the hypothesis and conclusion of the conditional statement. 1. The hypothesis of this statement is "a person is stealing"; the conclusion is "he is breaking the law." The negation of the hypothesis is "a person is not stealing." The inverse is "if a person is not stealing, he is not breaking the law." The inverse is false, since there are more ways to break the law than by stealing. Clearly, a murderer may not be stealing but he is surely breaking the law. 171 J.

CLEP College Mathematics

2. In this statement, the hypothesis contains two conditions: a) the line is perpendicular to the segment; and b) the line intersects the seg­ ment at the midpoint. The negation of (statement a and statement b) is (not statement a or not statement b). Thus, the negation of the hypothesis is "The line is not perpendicular to the segment or it doesn't intersect the segment at the midpoint." The negation of the conclusion is "the line is not the perpendicular bisector of a segment." The inverse is "if a line is not perpendicular to the segment or does not intersect the segment at the midpoint, then the line is not the perpendicu­ lar bisector of the segment." In this case, the inverse is true. If either ofthe conditions holds (the line is not perpendicular; the line does not intersect at the midpoint), then the line cannot be a perpendicular bisector. 3. This statement is not written in if-then form, which makes its hypothesis and conclusion more difficult to see. The hypothesis is implied to be "the man is dead"; the conclusion is implied to be "the man tells no tales." The inverse is, therefore, "If a man is not dead, then he will tell tales." The inverse is false. Many witnesses to crimes are still alive but they have never told their stories to the police, probably out of fear or because they didn't want to get involved.

NECESSARY AI

Let P and Q rE ment in which P is sary condition for 1

Example:

Consider the 5t it rains" is a sufficiE to the movies" is a r

Note that for tt dition for which Jan, Likewise, "Jane wil from a rainy weath~ another likely conel is a necessary eondi In the biconditi sufficient condition

BASIC PRINCIPLES, LAWS, AND THEOREMS Example:

I. Any statement is either true or false. (The Law of the Excluded Middle) 2. A statement cannot be both true and false. (The Law of Contradiction) 3. The converse of a true statement is not necessarily true. 4. The converse of a definition is always true. 5. For a theorem to be true, it must be true for all cases.

Consider the 5t, gets paid" is both a 51 Rick's working is a s Thus, we ha"e from the preceding s

7. The inverse of a true statement is not necessarily true.

11. If a given statem the hypothesis 01 conclusion of thi

8. The contrapositive of a true statement is true and the contrapositive of a false statement is false.

12. If a given statem are sufficient bUI

9. If the converse of a true statement is true, then the inverse is true. Like­ wise, if the converse is false, the inverse is false.

13. If a given statem are neither suffic

6. A statement is false if one false instance of the statement exists.

10. Statements that are either both true or both false are said to be logically equivalent.

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Chapter 8: Logic

mditions: a) the line e intersects the seg­ a and statement b) is rion of the hypothesis t doesn't intersect the :lusion is "the line is

NECESSARY AND SUFFICIENT CONDITIONS Let P and Q represent statements. "If P, then Q" is a conditional state­ ment in which P is a sufficient condition for Q, and similarly Q is a neces­ sary condition for P.

~ segment or does not s nor the perpendicu-

Example:

Consider the statement: "If it rains, then Jane will go to the movies." "If it rains" is a sufficient condition for Jane to go to the movies. "Jane will go to the movies" is a necessary condition for rain to have occurred.

Iltions holds (the line ~ midpoint), then the

Note that for the statement given, "Ifit rains" may not be the only con­ dition for which Jane goes to the movies; however, it is a sufficient condition. Likewise, "Jane will go to the movies" will certainly not be the only result from a rainy weather condition (for example, "the ground will get wet" is another likely conclusion). However, knowing that Jane went to the movies is a necessary condition for rain to have occurred.

makes its hypothesis is implied to be "the n tells no tales." The ill tell tales." ~ still alive but they It of fear or because

IS ~ Excluded Middle)

In the biconditional statement "P if and only if Q," P is a necessary and sufficient condition for Q, and vice versa.

I

Example: 11

II "1

\,

. of Contradiction)

,'I

,I

[Ii :,:

lie.

Consider the statement "Rick gets paid if and only if he works." "Rick gets paid" is both a sufficient and necessary condition for him to work. Also, Rick's working is a sufficient and necessary condition for him to get paid. Thus, we have the following basic principles to add to our list of ten from the preceding section:

rlent exists.

11. If a given statement and its converse are both true, then the conditions in the hypothesis of the statement are both necessary and sufficient for the conclusion of the statement.

:ontrapositive of a

12. If a given statement is true but its converse is false, then the conditions are sufficient but not necessary for the conclusion of the statement.

-erse is true. Like-

13. If a given statement and its converse are both false, then the conditions are neither sufficient nor necessary for the statement's conclusion.

aid to be logically

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CLEP College Mathematics

DEDUCTIVE REASONING An arrangement of statements that would allow you to deduce the third one from the preceding two is called a syllogism. A syllogism has three parts: 1. The first part is a general statement concerning a whole group. This is called the major premise. 2. The second part is a specific statement which indicates that a certain indi­ vidual is a member of that group. This is called the minor premise. 3. The last part of a syllogism is a statement to the effect that the general statement which applies to the group also applies to the individual. This third statement of a syllogism is called a deduction.

The flaw in this a condition on peop] are interchanged, thl

In the followin: tences, and develop linking them with th as operations transfi: describe them in gre, will find that differc different references.

TRUTH TABLES Example:

The truth possible Ie

This is an example of a properly deduced argument.

The logica and false (

A. Major Premise: All birds have feathers.

B. Minor Premise: An eagle is a bird. C. Deduction: An eagle has feathers.

The technique of employing a syllogism to arrive at a conclusion is called deductive reasoning. If a major premise that is true is followed by an appropriate minor premise that is true, a conclusion can be deduced that must be true, and the reasoning is valid. However, if a major premise that is true is followed by an inappropriate minor premise that is also true, a conclusion cannot be deduced.

Negation

If X is a sentenci contradiction of X T where ~ is called the

Example:

This is an example of an improperly deduced argument. A. Major Premise: All people who vote are at least 18 years old.

B. Improper Minor Premise: Jane is at least 18. C. Illogical Deduction: Jane votes.

174

Example:

For X = "Jane is ~ X = "Jane is

Chapter 8: Logic

ou to deduce the third , syllogism has three whole group. This is

res that a certain indi­ minor premise.

ffect that the general ) the individual. This

The flaw in this example is that the major premise in statement A makes a condition on people who vote, not on a person's age. If statements Band C are interchanged, the resulting three-part deduction would be logical. In the following we will use capital letters X, Y, Z, ... to represent sen­ tences, and develop algebraic tools to represent new sentences formed by linking them with the above connectives. Our connectives may be regarded as operations transforming one or more sentences into a new sentence. To describe them in greater detail, we introduce symbols to represent them. You will find that different symbols representing the same idea may appear in different references.

TRUTH TABLES AND BASIC LOGICAL OPERATIONS The truth table for a sentence X is the exhaustive list of possible logical values of X.

It.

The logical value of a sentence X is true (or and false (or F) if X is false.

~

at a conclusion is

appropriate minor lUst be true, and the true is followed by nclusion cannot be

l

n if X is true,

Negation If X is a sentence, then ~X represents the negation, the opposite, or the contradiction of X. Thus, the logical values of ~ X are as shown in Table 1, where ~ is called the negation operation on sentences. Table 1.-Truth Table for Negation

T

F

F

T

lent.

ars old. Example:

For X ~X

= "Jane is eating an apple," we have

= "Jane is not eating an apple."

175

-------t~------~-~-

_C_L_E_P_C_o_"_eg_e_M_a_th_e_m_a_t_ic_s

The negation operation is called unary, transforming a sentence into a unique image sentence.

IFF We use the symbol IFF to represent the expression "if and only if."

AND For sentences X and Y, the conjunction "X AND Y," represented by X A Y, is the sentence that is true IFF both X and Yare true. The truth table for A (or AND) is shown in Table 2, where A is called the conjunction operator.

Ii

I

As with the cc tion, transforming rl XVY.

Table 2-Truth Table for AND

Example:

For X = "Jane have X V Y = "Jane

T

T

T

T

F

F

F

T

F

IF-THEN

F

F

F

For sentences .\ "IF X THEN Y." .r ­ true. The truth table

The conjunction A is a binary operation, transforming a pair of sen­ tences into a unique image sentence.

cation operator.

Example:

For X = "Jane is eating an apple" and Y = "All apples are sweet," we have X A Y = "Jane is eating an apple AND all apples are sweet."

AND/OR For sentences X and Y, the disjunction "X AND/OR Y," represented by X V Y, denotes the sentence that is true if either or both X and Yare true. The truth table for V is shown in Table 3, where V is called the disjunction operator.

176

Implication is a and Y into a unique iT

Chapter 8: Logic

mg a sentence into a

Table 3-Truth Table for AND/OR

"if and only if."

T

T

T

T

F

T

F

T

T

F

F

F

. y.,. represented by

are true. The truth lIed the conjunction

ling a pair of sen-

Ies are sweet," we ~ sweet."

f':' represented by X and Yare true.

ed the disjunction

As with the conjunction operator, the disjunction is a binary opera­ tion, transforming the pair of sentences X, Y into a unique image sentence XVY.

Example:

For X = "Jane is eating the apple" and Y = "Marvin is running," we have X V Y = "Jane is eating the apple AND/OR Marvin is running."

IF-THEN For sentences X and Y, the implication X ---+ Y represents the statement "IF XTHEN Y." X ---+ Y is false IFF X is true and Y is false; otherwise, it is true. The truth table for ---+ is shown in Table 4. -~ is referred to as the impli­ cation operator. Table 4-Truth Table for IF-THEN

I

T

I

T

I

T

I

T

F

F

F

T

T

F

F

T

Implication is a binary operation, transforming the pair of sentences X and Y into a unique image sentence X ---+ Y.

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CLEP College Mathematics

LOGICAL EQUIVALENCE

Fundamental F

For sentences X and Y, the logical equivalence X - Y is true IFF X and Y have the same truth value; otherwise, it is false. The truth table for - is

The next three duced in another fo: number system.

shown by Table 5, where - represents logical equivalence, "IFF." Table S-Truth Table for Equivalence

THEOREM 2-Pro~

T

T

T

T

F

F

F

T

F

F

F

T

For any sentenci 1. Commutativity: X 2. Associativity: X .

THEOREM 3-Prop

For any sentence Example:

1. Commutativity: X

For X = "Jane eats apples" and Y = "apples are sweet," we have X - Y = "Jane eats apples IFF apples are sweet."

2. Associativity: X .

Equivalence is a binary operation, transforming pairs of sentences X and Y into a unique image sentence X - Y. The two sentences X, Y for which X - Yare said to be logically equivalent.

THEOREM 4-Distri

For any sentence 1. Xv (Y I\Z) - (X

Logical Equivalence versus "Meaning the Same"

2. X 1\ (Yv Z) -

(X

Logical equivalence (-) is not the same as an equivalence of mean­ ings. Thus, if Jane is eating an apple and Barbara is frightened of mice, then for X = "Jane is eating an apple" and Y = "Barbara is frightened of mice," X and Yare logically equivalent, since both are correct. However, they do not have the same meaning. Statements having the same meaning are, for example, the double negative ~ ~ X (not-not) and X itself.

2. ~(Xv Y) -

THEOREM i-Double Negation Equals Identity

Proof of Part 1 of Th

For any sentence X, ~~X-X

178

THEOREM 5-DeM(

For any

sentence~

1. ~(X 1\ Y) - (~XI (~.o':n

We can prove ~L over all possible com! assumed by the senten expression ~(X 1\ Y) iJ

Chapter 8: Logic

Fundamental Properties of Operations

Y is true IFF X and e truth table for - is ence, "IFF."

'-4

The next three theorems will look familiar because they were intro­ duced in another format in Chapter 2 for sets and Chapter 3 for the real number system.

THEOREM 2-Properties of Conjunction Operation

For any sentences X, Y, Z, the following properties hold: 1. Commutativity: X /\ Y -

Y /\ X

2. Associativity: X /\ (Y /\ Z) - (X /\ Y) /\ Z

THEOREM 3-Properties of Disjunction Operation

For any sentences X, Y, Z, the following properties hold: 1. Commutativity: Xv Y -

YV X

eet," we have X - Y

2. Associativity: Xv (YV Z) - (XV Y) V Z

airs of sentences X ~nces X, Y for which

THEOREM 4-Distributive Laws

~

For any sentences X, Y, Z, the following laws hold: 1. Xv (Y /\Z) -

Same"

Jiyalence of mean­

[ened of mice, then

ightened of mice,"

However, they do

e meaning are, for

f

(XV Y) /\ (XV Z)

2. X /\ (YV Z) - (X /\ Y) V (X /\ Z)

THEOREM 5-DeMorgan's Laws for Sentences

For any sentences X, Y, the following laws hold: 1.

~(X /\

Y) -

(~X)

V (~Y).

2.

~(XV

Y) -

(~X)

/\

(~Y).

Proof of Part 1 of Theorem 5:

We can prove ~(X /\ Y) - (~X) V (~ Y) by developing a truth table over all possible combinations of X and Y and observing that all values assumed by the sentences are the same. To this end, we first evaluate the expression ~(X /\ Y) in Table 6a.

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CLEP College Mathematics

Table 6a-Truth Table for Negation of Conjunction

Table 7i

y T

T

T

F

T

T

T

F

F

T

T

F

F

T

F

T

F

T

F

F

F

T

F

F

Now we evaluate ( ~ X) V ( ~ Y) in Table 6b.

For any sentenci equivalent, i.e.,

Table 6b-Truth Table for Disjunction of Negation ........................

This is pro\"en il

Table

T T

The last columns of the truth tables coincide, proving our assertion.

F

THEOREM 6-Two Logical Identities

F

For any sentences X, Y, the sentences X and (X A Y) V (X A logically equivalent. That is, (X A Y) V (X A

~

Y)

+-+

X

~ Y)

are

For any sentence

This is pro\"en in

This is proven in Table 7a.

f T

F F

180

Chapter 8: Logic

Ijunction

Table 7a-Truth Table for (X /\ Y) V (X /\ ~ Y)