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Table of contents
  1. Preface
  2. 1 Sets
    1. Introduction
    2. 1.1 Basic Set Concepts
    3. 1.2 Subsets
    4. 1.3 Understanding Venn Diagrams
    5. 1.4 Set Operations with Two Sets
    6. 1.5 Set Operations with Three Sets
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  3. 2 Logic
    1. Introduction
    2. 2.1 Statements and Quantifiers
    3. 2.2 Compound Statements
    4. 2.3 Constructing Truth Tables
    5. 2.4 Truth Tables for the Conditional and Biconditional
    6. 2.5 Equivalent Statements
    7. 2.6 De Morgan’s Laws
    8. 2.7 Logical Arguments
    9. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  4. 3 Real Number Systems and Number Theory
    1. Introduction
    2. 3.1 Prime and Composite Numbers
    3. 3.2 The Integers
    4. 3.3 Order of Operations
    5. 3.4 Rational Numbers
    6. 3.5 Irrational Numbers
    7. 3.6 Real Numbers
    8. 3.7 Clock Arithmetic
    9. 3.8 Exponents
    10. 3.9 Scientific Notation
    11. 3.10 Arithmetic Sequences
    12. 3.11 Geometric Sequences
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  5. 4 Number Representation and Calculation
    1. Introduction
    2. 4.1 Hindu-Arabic Positional System
    3. 4.2 Early Numeration Systems
    4. 4.3 Converting with Base Systems
    5. 4.4 Addition and Subtraction in Base Systems
    6. 4.5 Multiplication and Division in Base Systems
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  6. 5 Algebra
    1. Introduction
    2. 5.1 Algebraic Expressions
    3. 5.2 Linear Equations in One Variable with Applications
    4. 5.3 Linear Inequalities in One Variable with Applications
    5. 5.4 Ratios and Proportions
    6. 5.5 Graphing Linear Equations and Inequalities
    7. 5.6 Quadratic Equations with Two Variables with Applications
    8. 5.7 Functions
    9. 5.8 Graphing Functions
    10. 5.9 Systems of Linear Equations in Two Variables
    11. 5.10 Systems of Linear Inequalities in Two Variables
    12. 5.11 Linear Programming
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  7. 6 Money Management
    1. Introduction
    2. 6.1 Understanding Percent
    3. 6.2 Discounts, Markups, and Sales Tax
    4. 6.3 Simple Interest
    5. 6.4 Compound Interest
    6. 6.5 Making a Personal Budget
    7. 6.6 Methods of Savings
    8. 6.7 Investments
    9. 6.8 The Basics of Loans
    10. 6.9 Understanding Student Loans
    11. 6.10 Credit Cards
    12. 6.11 Buying or Leasing a Car
    13. 6.12 Renting and Homeownership
    14. 6.13 Income Tax
    15. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  8. 7 Probability
    1. Introduction
    2. 7.1 The Multiplication Rule for Counting
    3. 7.2 Permutations
    4. 7.3 Combinations
    5. 7.4 Tree Diagrams, Tables, and Outcomes
    6. 7.5 Basic Concepts of Probability
    7. 7.6 Probability with Permutations and Combinations
    8. 7.7 What Are the Odds?
    9. 7.8 The Addition Rule for Probability
    10. 7.9 Conditional Probability and the Multiplication Rule
    11. 7.10 The Binomial Distribution
    12. 7.11 Expected Value
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  9. 8 Statistics
    1. Introduction
    2. 8.1 Gathering and Organizing Data
    3. 8.2 Visualizing Data
    4. 8.3 Mean, Median and Mode
    5. 8.4 Range and Standard Deviation
    6. 8.5 Percentiles
    7. 8.6 The Normal Distribution
    8. 8.7 Applications of the Normal Distribution
    9. 8.8 Scatter Plots, Correlation, and Regression Lines
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  10. 9 Metric Measurement
    1. Introduction
    2. 9.1 The Metric System
    3. 9.2 Measuring Area
    4. 9.3 Measuring Volume
    5. 9.4 Measuring Weight
    6. 9.5 Measuring Temperature
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  11. 10 Geometry
    1. Introduction
    2. 10.1 Points, Lines, and Planes
    3. 10.2 Angles
    4. 10.3 Triangles
    5. 10.4 Polygons, Perimeter, and Circumference
    6. 10.5 Tessellations
    7. 10.6 Area
    8. 10.7 Volume and Surface Area
    9. 10.8 Right Triangle Trigonometry
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  12. 11 Voting and Apportionment
    1. Introduction
    2. 11.1 Voting Methods
    3. 11.2 Fairness in Voting Methods
    4. 11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem
    5. 11.4 Apportionment Methods
    6. 11.5 Fairness in Apportionment Methods
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  13. 12 Graph Theory
    1. Introduction
    2. 12.1 Graph Basics
    3. 12.2 Graph Structures
    4. 12.3 Comparing Graphs
    5. 12.4 Navigating Graphs
    6. 12.5 Euler Circuits
    7. 12.6 Euler Trails
    8. 12.7 Hamilton Cycles
    9. 12.8 Hamilton Paths
    10. 12.9 Traveling Salesperson Problem
    11. 12.10 Trees
    12. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  14. 13 Math and...
    1. Introduction
    2. 13.1 Math and Art
    3. 13.2 Math and the Environment
    4. 13.3 Math and Medicine
    5. 13.4 Math and Music
    6. 13.5 Math and Sports
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  15. A | Co-Req Appendix: Integer Powers of 10
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index

Key Concepts

2.1 Statements and Quantifiers

  • Logical statements have the form of a complete sentence and make claims that can be identified as true or false.
  • Logical statements are represented symbolically using a lowercase letter.
  • The negation of a logical statement has the opposite truth value of the original statement.
  • Be able to
    • Determine whether a sentence represents a logical statement.
    • Write and translate logical statements between words and symbols.
    • Negate logical statements, including logical statements containing quantifiers of all, some, and none.

2.2 Compound Statements

  • Logical connectives are used to form compound logical statements by using words such as and, or, and if …, then.
  • A conjunction is a compound logical statement formed by combining two statements with the words “and” or “but.” If the two independent clauses are represented by pp and qq, respectively, then the conjunction is written symbolically as pqpq. For the conjunction to be true, both pp and qq must be true.
  • A disjunction joins two logical statements with the or connective. In, logic or is inclusive. For an or statement to be true at least one statement must be true, but both may also be true.
  • A conditional statement has the form if pp, then qq, where pp and qq are logical statements. The only time the conditional statement is false is when pp is true, and qq is false.
  • The biconditional statement is formed using the connective if and only ifif and only if for the biconditional statement to be true, the true values of pp and qq, must match. If pp is true then qq must be true, if pp is false, then qq must be false.
  • Translate compound statements between words and symbolic form.
    Connective Symbol Name
    and
    but
    conjunction
    or disjunction, inclusive or
    not ~ negation
    if , then implies conditional, implication
    if and only if biconditional
  • The dominance of connectives explains the order in which compound logical statements containing multiple connectives should be interpreted.
  • The dominance of connectives should be applied in the following order
    • Parentheses
    • Negations
    • Disjunctions/Conjunctions, left to right
    • Conditionals
    • Biconditionals
    A table with four columns shows Dominance, Connective, Symbol, and Evaluate. The dominance column on the table shows a downward vertical arrow from least dominant to most dominant. The connective column on the table shows Parentheses, Negation, Disjunction or Conjunction, Conditional, and Biconditional. The Symbol column on the table shows an open bracket and a closed bracket, equivalent, an upward circumflex and a downward circumflex, a right side arrow, and a double-sided arrow. The Evaluate column on the table shows First, a downward arrow, Left to right or add parentheses to specify order because or slash and have equal dominance. a downward arrow, and last.
    Figure 2.18

2.3 Constructing Truth Tables

  • Determine the true values of logical statements involving negations, conjunctions, and disjunctions.
    • The negation of a logical statement has the opposite true value of the original statement.
    • A conjunction is true when both pp and qq are true, otherwise it is false.
    • A disjunction is false when both pp and qq are false, otherwise it is true.
  • Know how to construct a truth table involving negations, conjunctions, and disjunctions and apply the dominance of connectives to determine the truth value of a compound logical statement containing, negations, conjunctions, and disjunctions.
    Negation Conjunction (AND) Disjunction (OR)
    pp ~p~p pp qq pqpq pp qq pqpq
    T F T T T T T T
    F T T F F T F T
    F T F F T T
    F F F F F F
  • A logical statement is valid if it is always true. Know how to construct a truth table for a compound statement and use it to determine the validity of compound statements involving negations, conjunctions, and disjunctions.

2.4 Truth Tables for the Conditional and Biconditional

  • The conditional statement, if pp then qq, is like a contract. The only time it is false is when the contract has been broken. That is, when pp is true, and qq is false.
    Conditional
    pp qq pqpq
    T T T
    T F F
    F T T
    F F T
  • The biconditional statement, pp if and only if qq, it true whenever pp and qq have matching true values, otherwise it is false.
    Biconditional
    pp qq pqpq
    T T T
    T F F
    F T F
    F F T
  • Know how to construct truth tables involving conditional and biconditional statements.
  • Use truth tables to analyze conditional and biconditional statements and determine their validity.

2.5 Equivalent Statements

  • Two statements pp and qq are logically equivalent if the biconditional statement, pqpq is a valid argument. That is, the last column of the truth table consists of only true values. In other words, pqpq is a tautology. Symbolically, pp is logically equivalent to qq is written as: pq.pq.
  • A logical statement is a tautology if it is always true.
  • To be valid a local argument must be a tautology. It must always be true.
  • Know the variations of the conditional statement, be able to determine their truth values and compose statements with them.
  • The converse of a conditional statement, if pp then qq, is the statement formed by interchanging the hypothesis and conclusion. It is the statement if qq then pp.
  • The inverse of a conditional statement if formed by negating the hypothesis and the conclusion of the conditional statement.
  • The contrapositive negates and interchanges the hypothesis and the conclusion.
    Conditional Contrapositive Converse Inverse
    pp qq ~p~p ~q~q pqpq ~q~p~q~p qpqp ~p~q~p~q
    T T F F T T T T
    T F F T F F T T
    F T T F T T F F
    F F T T T T T T
  • The conditional statement is logically equivalent to the contrapositive.
  • The converse is logically equivalent to the inverse.
  • Know how to construct and use truth tables to determine whether statements are logically equivalent.

2.6 De Morgan’s Laws

  • De Morgan’s Law for the negation of a disjunction states that, ~(pq)~(pq) is logically equivalent to ~p~q.~p~q.
  • De Morgan’s Law The negation of a conjunction states that, ~(pq)~p~q.~(pq)~p~q.
  • Use De Morgan’s Laws to negate conjunctions and disjunctions.
  • The negation of a conditional statement, if pp then qq is logically equivalent to the statement pp and not qq. Use this property to write the negation of conditional statements.
  • Use truth tables to evaluate De Morgan’s Laws.

2.7 Logical Arguments

  • A logical argument uses a series of facts or premises to justify a conclusion or claim. It is valid if its conclusion follows from the premises, and it is sound if it is valid, and all of its premises are true.
  • The law of detachment is a valid form of a conditional argument that asserts that if both the conditional, pqpq is true and the hypothesis, pp is true, then the conclusion qq must also be true.
    Law of Detachment
    Premise: pqpq
    Premise: pp
    Conclusion: q q
  • Know how to apply the law of detachment to determine the conclusion of a pair of statements.
  • The law of denying the consequent is a valid form of a conditional argument that asserts that if both the conditional, pqpq is true and the negation of the conclusion, ~q~q is true, then the negation of the hypothesis ~p~p must also be true.
    Law of Denying the Consequent
    Premise: pqpq
    Premise: ~q~q
    Conclusion: ~p ~p
  • Know how to apply the law of denying the consequent to determine the conclusion for pairs of statements.
  • The chain rule for conditional arguments is a valid form of a conditional argument that asserts that if the premises of the argument have the form, pqpq and qrqr, then it follows that pr.pr.
    Chain Rule for Conditional Arguments
    Premise: pqpq
    Premise: qrqr
    Conclusion: pr pr
  • Know how to apply the chain rule to determine valid conclusions for pairs of true statements.
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