CMPSCI 250: Introduction to Computation

First Midterm Exam

David Mix Barrington

27 September 2007

Directions:

• Answer the problems on the exam pages.
• There are five problems on pages 2-6, for 100 total points plus 10 extra credit. Probable scale is somewhere around A=93, C=69.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.

  Q1: 20 points
  Q2: 20 points
  Q3: 20 points
  Q4: 20 points
  Q5: 20 points plus 10 extra credit
 Total: 100 points plus 10 extra credit

Questions 1, 2, and 3 deal with a set T of terriers including two specific terriers Angus (a) and Barney (b). If x is a terrier, the predicate C(x) means "x likes to chase squirrels" and D(x) means "x likes to dig". If x and y are two terriers, SS(x,y) means "x and y are the same size". You are given that SS is an equivalence relation.

• Question 1 (20): Translate the following statements as indicated:
  • Statement I: (to English) (C(b) ∧ D(a)) → (D(b) ∧ ¬D(b))
  • Statement II: (to symbols) If either Angus likes to chase squirrels or Barney likes to dig, or both, then Angus likes to dig.
  • Statement III: (to English) ¬((C(a) → C(b)) ∧ (C(b) → C(a)))
  • Statement IV: (to symbols) Given any terrier, there is a terrier of the same size that likes to dig.
  • Statement V: (to English) ∀x: D(x) → ¬C(x)
  • Statement VI: (to symbols) There are two terriers of the same size such that one likes to chase squirrels and the other doesn't.

• Question 2 (20): Using Statements I, II, III, and V from Question 1, determine whether each of the propositions C(a), C(b), D(a), and D(b) are true or false. Check that given your answers, Statements I, II, and III are true, and that V is true in so far as it applies to Angus and Barney. (You may use truth tables or a deductive sequence proof.)

• Question 3 (20): Using Statements I, II, III, IV, and V from Question 1, and any facts you were able to deduce from Question 2, prove Statement VI. Since both some of your premises and your conclusin involve quantifiers, you will need some or all of the quantifier rules of Existence, Instantiation, Specification, and Generalization.

• Question 4 (20): In this problem, Amy, Bob, and Claire are three students who each have zero or more majors taken from the set {Dance, English, French}. We write M(x,y) to mean "student x has major y" (possibly with other majors as well). You are given that M(a,d) and M(a,e) are false, and that M(b,d) and M(c,e) are true. Recall that a relation R from one set to another is a function if and only if it is both total (∀x:∃y: R(x,y)) and well-defined (∀x:∀y:∀z:[R(x,y) ∧ R(x,z)] → (y=z)).
  • (a,5) If we are given, in addition to the four truth values above, that M is a function, deterine the truth values of the five propositions M(x,y) that are not given above.
  • (b,5) If M is a function as in (a), is it a surjection (and onto function)? Is it an injection (a one-to-one function)? Explain your answer indicating that you understand the relevant definitions.
  • (c,5) Give truth values for the five propositions not given above that make M a total relation that is not well-defined. Explain why your values do this.
  • (d,5) Give truth values for the five propositions not given above that make M a well-defined relation that is not total. Explain why your values do this.

• Question 5 (20+10): Let X be an arbitrary set and let R and S be two arbitrary partial orders on X. That is, R is:
  • Reflexive: ∀x: R(x,x)
  • Antisymmetric: ∀x:∀y: [R(x,y) ∧ R(y,x)] → (x=y)
  • Transitive: ∀x:∀y:∀z: [R(x,y) ∧ R(y,z)] → R(x,z)

  Here are your questions:

  • (a,20) Let T be the binary relation defined by T(x,y) ↔ (R(x,y) ∧ S(x,y)) (that is, T is the intersection of R and S). Prove that T is a partial order by carefully verifying each of the three properties above using quantifier rules and the fact that R and S are partial orders.
  • (b,10 XC) Let U be the union of R and S, so that U(x,y) ↔ (R(x,y) ∨ S(x,y)). Is U necessarily a partial order? For each of the three properties, either prove that the property holds for U given any partial orders R and S, or indicate how it could fail for U on some particular partial orders R and S.

  Last modified 28 September 2007