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Math 101 Exam #1 – Study Guide We’ve covered Chapter 1 – LOGIC and Chapter 6 – VOTING and APPORTIONMENT. The test consists of 10 problems, plus a few bonus points possible. ALL PROBLEMS REQUIRE SENTENCE ANSWERS and many require you to explain your reasoning. Several of the problems have multiple parts to them. In short, study each chapter and be able to do everything we’ve covered in each one! Seriously! The test is pretty comprehensive and pretty long. I’ll be giving you two hours to work on the exam, followed by 1 hour on new material from Chapter 5. When I look through the chapters and the notes, here are some highlights that you should be able to reproduce on the day of the test. Chapter 1 – Logic Define and correctly use all the terms on page 58 Work through as many different types of Review exercises as you can, starting on page 58. Stick to ones like the ones we did in class. Skip review problems that were skipped in the homework (such as Sudoku). Verify the validity, or demonstrate the invalidity, of arguments using Venn diagrams. Describe the difference between inductive and deductive logic and identify whether a given argument is inductive or deductive. Set up symbolic notation for given phrases in English. Understand and work with conditionals, negations, “and” and “or” statements Know DeMorgan’s Laws and be able to prove them using either Venn diagrams or truth tables Write out in English the negation of a given phrase or argument. Know when a truth table is showing that an argument is true (Answer: When it’s a tautology – but be sure to know what that is!) Construct a truth table to verify the validity of an argument (like in section 1.5). There’s definitely one of these on the test. Chapter 6 – Voting and Apportionment Define and correctly use all the terms on page 473. Work through several of the Review Exercises starting on page 473. Determine the winner of an election using each of the voting methods we covered o Plurality o Plurality with elimination o Instant Runoff/Ranked Choice o Borda Count o Pairwise Comparison o Approval voting Distinguish the key differences between methods. Hardest one: “What’s the difference between the plurality with elimination method and the instant runoff method?” Describe the basic concept behind the four fairness criteria for voting o Majority Criterion o Head-to-head Criterion o Monotonicity Criterion o Irrelevant Alternatives Criterion State Arrows’ Impossibility Theorem and describe, in general terms, what it’s saying. Calculate the fundamental quantities for a given electoral situation o Standard divisor o Standard quota o Lower quota o Upper quota Determine the apportionment of a given number of seats using each method we’ve studied: o Hamilton’s Method o Jefferson’s Method o Adams’ Method o Webster’s Method o Hill-Huntington Method Know that all but Hamilton’s Method require a modified divisor. Correctly determine whether to choose a smaller or a larger modified divisor for each method – and explain why. State and describe the rationale behind the Quota Rule. Identify when a given situation violates the Quota Rule. State the three different apportionment paradoxes and identify when one is “rearing its ugly head” in a given problem o Alabama Paradox o New States Paradox o Population Paradox State and describe the Balinski-Young Impossibility Theorem Compare and contrast Arrow’s Impossibility Theorem with the Balinski-Young Impossibility Theorem. Create a list of similarities and differences and be prepared to author a paragraph comparing and contrasting the two.
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