Questions and Answers (1)

Here are questions with answers that you might help you to prepare for the midterm exam.

Questions regarding the second half of the course are here.


  1. What did the Egyptians use to construct a right angle?
  2. A cord with knots. (Lecture).

  3. Describe in a sentence the relation between Pythagoras and the Pytagorean theorem.
  4. Pythagoras (or one of his followers) stated the theorem in general and gave a proof for it. Implicitly it was well known already to the Egyptians and Babylonians. (Lecture).

  5. Who is the "first philosopher"?
  6. Thales of Miletus. (Lecture)

  7. What is the author's explanation for the fact that Chinese children can in average recite longer series of numbers than American children of the same age?
  8. The Chinese number system is superior: it is more systematic and there are less and shorter words for numbers. (Lecture).

  9. Say when the last ice age was, or why was it mentioned in class.
  10. It was mentioned because around that time people started to settle, thus having more time and more needs for mathematics. The last ice age was 12000-10000 B.C. (Lecture)

  11. When are two lengths commensurable?
  12. Two lenghts AB and CD are commensurable if they can be measured by the same unit. Formally: if there exists a rational number u and natural numbers x and y, such that u*x=AB and u*y= CD. (Lecture).

  13. What is so special about 60 that it has become the base of many number systems?
  14. 60 has many divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60); it is also related to early calendars (with 360 days); but, ultimately, we do not know. (Lecture).

  15. When is an argument deductively valid?
  16. An argument is deductively valid, if it is impossible for the premises to be true and for the conclusion to be false at the same time (Lecture; FOL, p.24).

  17. What is an inductive argument?
  18. An argument where the truth of the premises make the conclusion more likely to be true (Lecture).

  19. What is the general structure of an argument of the form of modus ponens?
  20. A          A->B

    (Handout #2: Weston, page 47). 

  21. When is a judgment objective (within a certain community)?
  22. A judgment is objective if its outcome does not depend on the judge; taste of coffee vs. validity of argument. (Lecture).

  23. What is a formal argument?
  24. An argument is formal if the validity of the argument does not depend on the meaning of the symbols employed. (Handout #1).

  25. What is the structure of an argument of the form reductio ad absurdum?
  26.   not A

    From the assumption `not A' and the derivation of a contradiction you infer `A'. (Lecture)

  27. If you know that the premises of an argument are true, and the conclusion is true, what do you know about the argument? (Is is valid?)
  28. You know nothing, the argument could as well be invalid. (Lecture)

  29. Consider the following argument:
      If taxes are lowered, my income rises.
      My income rises.
      Therefore, my taxes are lowered.
    Assuming that all three statements are true, is this a valid argument? Justify your answer in one sentence.
  30. The argument is not valid. Because it is possible for the premises to be true, but the conclusion to be false, for example, if my income rises because of a raise and not because of the lowering of my taxes. (Lecture)

  31. What is a sound argument?
  32. An argument is sound if it is valid and its premises are true (Lecture).

  33. What entities can be true or false?
  34. Propositions (statements, declarative sentences) (Lecture).

  35. Name a theory (other than Geometry) that has been axiomatized.
  36. Number theory, set theory, utility theory, physics, and many more (Lecture).

  37. When is an axiom system consistent?
  38. If it is not possible to derive a contradiction from the axioms (Lecture).

  39. If I say "X is valid", what does X have to be? (Statement, axiom system, proposition, argument, etc.)
  40. An argument. (Lecture).

  41. When is an axiom independent of other axioms?
  42. An axiom of a consistent axioms system is independent if its negation together with the remaining axioms is still consistent. Another way to say this is that the axiom cannot be proved from the other axioms. (Lecture).

  43. Is the axiom system consisting of Euclid's first four axioms and the negation of his fifth axiom consistent?
  44. Yes. One can present models for Non-Euclidean geometries. (Lecture).

  45. In what century where the axioms for arithmetic formulated?
  46. In the 19th century: 1888 by Dedekind, and 1889 by Peano. (Lecture).

  47. In Tarski's World, can an object have more than one name? How about in our (the actual, real) world?
  48. Yes in both cases. (Lecture).

  49. State the three axioms of probability theory.
  50. 1. 0 <= P(A) <= 1
    2. P(True)=1
    3. P(A or B) = P(A)+P(B), when A and B cannot both be true at the same time.
    (Lecture, Handout #10).

  51. State either the definition of Conditional Probability or Bayes' Theorem.
  52. Conditional Probability: P(A|B) = P(A and B) / P(B)
    Bayes' Theorem: P(A|B) = ( P(B|A)*P(A) ) / P(B)
    (Lecture, Handout #10).

  53. Assume you woke up this morning and had no idea whatsoever about what day of the week it was. It could be any of { Mon, Tue, Wed, Thu, Fri, Sat, Sun }. If P(Thu) stands for "the probability that today is Thursday", what is the value of P(Thu)?
  54. P(Thu) = 1/7

  55. Let 80-110 stand for "there's 80-110 lecture today". What is the value of P( Thu | 80-110 )?
  56. P( Thu | 80-110 ) = 0, there is no class on Thursdays.

© Dirk Schlimm, Last modified: 3/1/02