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(Section 0.1: Sets of Numbers) 0.1.2 , or Z, the set of integers. This set consists of 1, 2, 3, etc.; their opposites, 1 , 2 , 3 , etc.(making up , the set of negative integers); and 0. , or Q, the set of rational numbers. This is the set of all numbers that can be written in the form:integernonzero integer It is the set of numbers that can be written as finite (or terminating)decimals or repeating decimals. Examples include:Fraction form1213823 99007, or 71Decimal form0.50.3, or 0.3333 0.831, or 0.8313131 7 As demonstrated by the last example, every integer is a rationalnumber. That is, . comes from “Quotient.” , or R, the set of real numbers. This is the set of all numbers that can be written as decimals. This set can be represented by the real number line below.There should be tick marks at 0 and at least one other number toindicate scale. This set consists of rational numbers and irrational numbers, suchas π and2.

(Section 0.1: Sets of Numbers) 0.1.3 , or C, the set of complex numbers. This set consists of real numbers and imaginary numbers, such asi and 2 3i. (See Chapter 2.)The Venn diagram below indicates the (proper) subset relations: . Each disk is contained within each larger disk.PART C: SET NOTATIONSet Notation denotes “in,” or “is in.” It denotes set membership. denotes “is not in.” denotes “such that.” (the universal quantifier) denotes “for all” or “for every.” (the existential quantifier) denotes “there exists” or “there is.”Example Set 1 (Using Set Notation)7 1 2 x , a, b , x , x x 1 x , y x 2y§Plain English translation7 is an integer.1is not an integer.2For all real numbers x, For all integers a and b, Every real number is less than one added to itself.Every real number is twice some real number.

(Section 0.1: Sets of Numbers) 0.1.4PART D: INFINITYInfinity, denoted by the lemniscate , is a quantity that is greater than anyreal number.Negative infinity, denoted by , is a quantity that is lesser than anyreal number.WARNING 2: and are not real numbers, though they may be handleddifferently in higher math.PART E: REPRESENTING SETS OF REAL NUMBERSThere are various ways to represent a set of real numbers.We sometimes list the members of a set and surround them with braces.For example, the set 1, π consists of the two members 1 and π .{ }An interval corresponds to a connected (“unbroken”) piece of the real numberline.A bounded interval has finite length on the real number line.An unbounded interval has infinite length.An open interval excludes its endpoints.A closed interval includes its endpoints.Example 2 (A Bounded, Open Interval)The set of all real numbers x such that 3 x 5 in set-builder form is:{x }3 x 5 , or{ x : 3 x 5 } in graphical form is: in interval form is:(3, 5)The set is an interval with 3 and 5 as its endpoints.The set is an open interval, because it excludes its endpoints.The exclusion of 3 and 5 is indicated by the use of: strict inequality signs ( ) in set-builder form, hollow circles (or parentheses) in graphical form, and parentheses in interval form. §

(Section 0.1: Sets of Numbers) 0.1.5Example 3 (A Bounded, Closed Interval)The set of all real numbers x such that 3 x 5 in set-builder form is:{x }3 x 5 , or{ x : 3 x 5 } in graphical form is: in interval form is: 3, 5 The set is a closed interval, because it includes its endpoints.The inclusion of 3 and 5 is indicated by the use of: weak inequality signs ( ) in set-builder form, filled-in circles (or brackets) in graphical form, and brackets in interval form. §Example Set 4 (Various Types of Intervals)Set-builderForm{x 3 x 5 }{x 3 x 5 }{x x 3 }{x x 3 }{x x 3 }{x x 3 }{x x } ,which is GraphicalFormIntervalForm) 3, 5(3, 5 (3, ) 3, )( , 3)( , 3 ( , )Type dedUnboundedUnboundedUnboundedUnbounded§WARNING 3: In interval form, parentheses are always placed next to and , because they are not real numbers and are therefore excluded from the set.However, may be considered to be both an open interval and a closed interval.

(Section 0.1: Sets of Numbers) 0.1.6.A B , the union of set A and set B, consists of all elements in one or both sets.Think: “All members are invited.”{} A B x x A or x B , where “or” is taken to be “inclusive or,” or“and/or.” The disjunctive symbol denotes “or.”A B , the intersection (“overlap”) of set A and set B, consists of all elements inboth sets.{} A B x x A and x B . The conjunctive symbol denotes “and.”Example 5 (A Union of Intervals){ }The set \ 1, π , which is written in set-difference form, consists of all realnumbers except 1 and π . The set in set-builder form is:{x }x 1 and x π , or{ x : x 1 andx π } in graphical form is: in interval form is:( , 1) (1, π ) (π , )TIP 1: The graphical form can help us write the interval form.TIP 2: The union symbol separates intervals in interval form whenevera real number must be “skipped over.” §Example 6 (Union and Intersection of Intervals)( )( )))a) Simplify 1, 5 3, . (Think: “All members are invited.”)b) Simplify 1, 5 3, . (Think: “Overlap.”)§ SolutionIntervalForm(1, 5) 3, )( )) ( )b) (1, 5) 3, ) 3, 5)a) 1, 5 3, 1, §GraphicalForm

(Section 0.2: Logic) 0.2.1SECTION 0.2: LOGICLEARNING OBJECTIVES Be able to identify and use logical notation and terminology. Understand the structure of “if-then” and “if and only if” statements. Understand counterexamples and logical equivalence. Know how to find the converse, inverse, and contrapositive of an “if-then” statement. Understand necessary conditions and sufficient conditions.PART A: DISCUSSION Although logic is a subject that is often relegated to courses in discretemathematics, computer science, and electrical engineering, its fundamentals areessential for clear and precise mathematical thought. Many theorems are “if-then” or “if and only if” statements.PART B: PROPOSITIONS AND “IF-THEN” STATEMENTSA proposition is a statement that is either true or false.“If-Then” StatementsThe statement “If p, then q” can be written as “ p q .” The proposition p is called the hypothesis, assumption, or condition. The proposition q is called the conclusion. If there are no cases where p is true and q is false, we say that thestatement is true. Otherwise, the statement is false, and any case where p is true andq is false is called a counterexample.If the statement is known to be true, we can write “ p q .”“ ” may be read as “implies.” WARNING 1: “ ” denotes “approaches” when we discuss limits incalculus.

(Section 0.2: Logic) 0.2.2Example 1 (An “If-Then” Statement)Consider the statement: “If I get an A, then I pass.”This statement is of the form “If p, then q,” where:p is the hypothesis “I get an A,” andq is the conclusion “I pass.”The statement is true, because there is no case where a student gets an A butdoes not pass. §The converse of p q is q p .Example 2 (An “If-Then” Statement and Its Converse)Consider the statement:“If a number is an integer, then it is a rational number.” That is, ( x ) ( x ) .The statement is true, because every integer is also a rational number.We can write: ( x ) ( x ) .Now, consider the converse of the above statement:“If a number is a rational number, then it is an integer.” That is, ( x ) ( x ) .This second statement is false, because a counterexample exists.1Observe that is a rational number, yet it is not an integer.2WARNING 2: The discovery of even one counterexample can beused to disprove a statement (that is, to prove that a statement isfalse). However, a single example is usually not enough to prove thata statement is true. To prove that a statement is true, one often needsto present a rigorous and general argument that applies to all caseswhere the hypotheses hold.Because the converse is false, we can write: ( x ) ( x ) ,where “ ” denotes “does not imply.” §

(Section 0.2: Logic) 0.2.3PART C: “IF AND ONLY IF” (or “IFF”) STATEMENTS“If and Only If ” (or “Iff”) StatementsThe statement “p if and only if q,” or “p iff q,” can be written as “ p q .”If the statement is known to be true, we can write “ p q ,” or “ p q .” If p q , and if q p , then p q . Then, the propositions p and q are logically equivalent;either both are true, or both are false.() () For example, 2x 6 x 3 . (See Footnote 1.) Definitions are essentially “iff” statements. For example, a number is a rationalintegernumber iff it can be written in the form.nonzero integer True “iff” statements arise when an “if-then” statement and its converse are true.PART D: CONVERSE, INVERSE, AND CONTRAPOSITIVEThe converse of p q is:The inverse of p q is:q p p q “ ” and “ ” are used to denote “not.” They are negation operators.The contrapositive of p q is: q p TIP 1: Take the original statement, switch the propositions, andnegate them.Contrapositive TheoremIf an “if-then” statement is true, then its contrapositive must be true, andvice-versa. In other words, they are logically equivalent.() ()That is, p q q p . This can be proven using truth tables in a discrete mathematics class.As a result, any “if-then” statement you know to be true has acontrapositive associated with it that will automatically be true.WARNING 3: An “if-then” statement may or may not be logically equivalent toits converse or its inverse. (However, the converse and the inverse must belogically equivalent to each other. Why? How are they related?)

(Section 0.2: Logic) 0.2.4Example 3 (Converse, Inverse, and Contrapositive of an “If-Then” Statement;Revisiting Example 1)Consider the true “if-then” statement: If I get an A , then I pass . pqWhat are its converse, inverse, and contrapositive? Which of these are true?§ SolutionConverse:If I pass , then I get an A . qpThis statement is false, because we can find a counterexample.A student can pass the class with a grade of B or C.Inverse:If I do not get an A , then I do not pass . p qThis statement is false, because we can find a counterexample.A student can have a grade of B or C, and the student can pass.Contrapositive: If I do not pass , then I do not get an A . q pBy the Contrapositive Theorem, this must be true, since the originalstatement is.The Venn diagram below may clarify matters. (We ignore “ ” and “ ”modifiers, as well as pass/fail grading.)§

(Section 0.2: Logic) 0.2.5.Example 4 (An “Iff” Statement)Consider the (true) statement:If I get an A, B, or C , then I pass . pqIts converse is true:If I pass , then I get an A, B, or C . qpTherefore, this is true:I get an A, B, or C iff I pass . § pqPART E: NECESSARY CONDITIONS and SUFFICIENT CONDITIONS(p is a necessary condition for q q p) That is, for q to be true, it is required that p must be true.(p is a sufficient condition for q p q) That is, if p is true, then it is guaranteed that q must be true.Example 5 (Necessary Conditions and Sufficient Conditions; Revisiting Example 1)Consider the “if-then” statement: If I get an A , then I pass . pqIs p sufficient for q? Is p necessary for q?§ Solutionp is sufficient for q, because p q . (Equivalently, q is necessary for p.)Think: If I get an A, then I am guaranteed to pass.p is not necessary for q, because q p .Think: I don’t need to get an A to pass. §Example 6 (“Necessary and Sufficient” Conditions; Revisiting Example 4)Consider the “iff ” statement: I get an A, B, or C iff I pass . pqThis is true, so p q , and we say that p is necessary and sufficient for q.Therefore, p and q are logically equivalent. §FOOTNOTES1. Propositions. Unlike some sources such as Rosen’s Discrete Mathematics and Its Applications,we call “ 2x 6 ” a proposition, even though its truth value (“true” or “false”) depends on thevalue of x.

(Section 0.3: Rounding) 0.3.1SECTION 0.3: ROUNDINGLEARNING OBJECTIVES Be able to round off decimals to a specified decimal place, a specified number ofdecimal places, or a specified number of significant digits (or figures). Understand scientific notation.PART A: DISCUSSION (WARNINGS and TIPS) Using CalculatorsMemory buttons on calculators can help preserve decimal accuracy.Taking too few decimal places may ruin the accuracy of final answers.A calculator might approximate π as 3.14159265. Though seeminglyimpressive, this is not an exact representation of π . Calculators cannotgive exact decimal representations of such numbers as 2 , ln5 , and sin 37 . Exact vs. Approximate AnswersMath instructors typically expect exact answers, though decimalapproximations may be required for “word problems.”For example, if an answer is π , you should write π .You may also be asked to give a decimal approximation such as 3.14. “Word Problems”Depending on context, decimal answers might need to be rounded down,rounded up, or rounded off.Write any appropriate units such as feet, pounds, etc. as part of your answer.PART B: ROUNDING“ ” means “is approximately” or “is about.”Example 1 (Rounding Off to a Specified Decimal Place)π 3.14159 . π rounded off to the nearest integer is:3 tenth (that is, to one decimal place) is:3.1 hundredth (that is, to two decimal places) is:3.14 thousandth (that is, to three decimal places) is: 3.142 In the U.S., the “1” in the third decimal place of π is rounded up to “2.”This is because the digit in the next decimal place is 5 or higher.(Different countries have different rules if that digit is a “5.”) §

(Section 0.3: Rounding) 0.3.2.Consider a number written in decimal form.To count decimal places, count all digits to the right of the decimal point.To count significant digits (or figures), locate the leftmost nonzero digit, andcount it and all digits to the right of it. We may want numbers written out to the same number of decimal places whenadding and subtracting them, while we may focus on significant digits whenmultiplying and dividing them. We can also use significant digits to roughlyretain the same level of accuracy when handling different units such as inches,feet, and miles.Example 2 (Counting Decimal Places and Significant Digits)Consider: 70.1230 This is written out to four decimal places.We count the four digits after the decimal point. This is written out to six significant digits.We include the two digits to the left of the decimal point. The “0” at the end indicates that we claim accuracy to four decimalplaces. Writing “70.123” might not have had that effect. We call 70 the integer part of this decimal. §Example 3 (Counting Decimal Places and Significant Digits; Scientific Notation)Consider: 0.001020, or .001020 (the integer part is 0; writing it is optional). These are written out to six decimal places. We count all six digitsafter the decimal point, including the two zeros after the decimal pointand before the “1.” These are written out to four significant digits. We do not includethe aforementioned zeros, but we include the “1” and all digits after it. These can be written as 1.020 10 3 using scientific notation.The “ 3 ” exponent tells us to move the decimal point three places tothe left to obtain .001020 in standard form. Note that:1.020 103 1020. 1020 , since the “3” exponent tells us to move thedecimal point three places to the right. §

(Section 0.4: Absolute Value and Distance) 0.4.1SECTION 0.4: ABSOLUTE VALUE AND DISTANCELEARNING OBJECTIVES Be able to formally define absolute value. Know how to take the absolute value of a real number. Know properties of the absolute value operation. Relate absolute value and distances on the real number line.PART A: DISCUSSION Understanding the definition of absolute value is crucial to understanding theabsolute value function and its graph. (See Section 1.3.)PART B: ABSOLUTE VALUEDefinition of Absolute ValueThe absolute value of a is denoted by a , where a, if a 0a a, if a 0 That is, the absolute value of a nonnegative real number is itself,while the absolute value of a negative real number is its opposite. This is an example of a piecewise definition because of the different rulesfor different values of a. (See Section 1.5.)TIP 1: The absolute value of a real number is never negative.WARNING 1: a does not necessarily represent a negative value. In fact,if a 0 , then a 0 . That is, the opposite of a negative real number is positive.Example 1 (Finding Absolute Value)a) Find 7 . b) Find 0 . c) Find 3 .§ Solutiona) 7 0 , so we use the top rule in the definition. 7 7 .b) 0 0 , so we again use the top rule in the definition. 0 0 .c) 3 0 , so we use the bottom rule in the definition: 3 ( 3) 3 . §

(Section 0.4: Absolute Value and Distance) 0.4.2PART C: PROPERTIES OF ABSOLUTE VALUEProperties of Absolute Value1) a 0Absolute values are never negative. (TIP 1)2) a aOpposites have the same absolute value.The absolute value of a product equals theproduct of the absolute values.The absolute value of a quotient equals thequotient of the absolute values, provided thedenominators are nonzero.3) ab a b4)aa , if b 0bbPART D: ABSOLUTE VALUE, DISTANCE, and OPPOSITESInterpreting Absolute Value as a Distancea the distance between a and 0 on the real number line.TIP 2: Just as for absolute value, a distance cannot be negative. However, we douse “signed distances” and “signed lengths” in trigonometry and motion problems.Example 2 (Interpreting Absolute Value as a Distance) 3 3 and 3 3 , because both 3 and 3 are at a distance of 3 unitsaway from 0 on the real number line.§The Distance Between a and bb a a b the distance between a and b on the real number line. In particular, a 0 a the distance between a and 0 on thereal number line. Absolute value notation allows us to denote such distances withoutworrying about whether a is greater than b or vice

quantities in calculus, particularly when defining limits. Notation Name Comments δ delta This is the fourth letter of the Greek alphabet. ε epsilon . needed in precalculus and calculus. Sections 0.2, 0.8, 0.9, 0.10, and 0.16 may be largely unfamiliar to incoming precalculus students.