CMPSCI 250: Introduction to Computation

First Midterm Exam Fall 2018

David Mix Barrington and Marius Minea

10 October 2018

Directions:

• Answer the problems on the exam pages.
• There are four problems, each with multiple parts, for 100 total points plus 10 extra credit. Actual scale A = 93, C = 63.
• Some useful definitions precede the questions below.
• No books, notes, calculators, or collaboration.
• In case of a numerical answer, an arithmetic expression like "2¹⁷ - 4" need not be reduced to a single integer.

  Q1: 20 points
  Q2: 30 points
  Q3: 30 points
  Q4: 20+10 points
Total: 100+10 points

Here are definitions of sets, predicates, and statements used on this exam.

Remember that the score of any quantifier is always to the end of the statement it is in.

Question 2 deals with the following scenario. One day Cardie and Duncan were joined on their morning walk by several other dogs. The set S of dogs on this group walk included Bingley (b), Cardie (c), Duncan (d), Guinness (g), Whistle (w), and perhaps others.

Let the binary predicate JB on S be defined so that JB(x, y) means "dog x joined the walk before dog y". Assume that the relation corresponding to JB is antireflexive, antisymmetric, and transitive.

Let N be the set of natural numbers {0, 1, 2, 3,...}.

If a, b, and m are naturals, with m > 0, the notation "a ≡ b (mod m)" means "a is congruent to b, modulo m".

The operator "%" on naturals, as in Java, refers to integer division, so that "x % y" is the remainder on dividing x by y.

• Question 1 (20): Translate each statement as indicated, using the set of dogs S which includes the dogs Bingley (b), Cardie (c), Duncan (d), Guinness (g), and Whistle (w), and possibly other dogs as well. All variables should be of type "dog". The predicate JB is defined so that JB(x, y) means "dog x joined the walk before dog y".
  • (a, 3) (to English) (Statement I) ∀x: JB(c, x) ↔ JB(d, x)

  • (b, 3) (to symbols) (Statement II) If Bingley joined the walk before Whistle, then Cardie joined before Guinness and Guinness joined before Bingley.

  • (c, 3) (to English) (Statement III) JB(g, b) ∧ (JB(b, w) ⊕ JB(c, g))

  • (d, 4) (to symbols) (Statement IV) Whistle joined the walk before at least two distince other dogs.

  • (e, 4) (to symbols) (Statement V) Every dog, except for Bingley himself, joined the walk before Bingley.

  • (f, 3) (to English) (Statement VI) ¬(∀y: JB(y, w) ∨ JB(b, y))

• Question 2 (30): These questions use the definitions, predicates, and premises above.
  • (a, 10) Assume that Statements II and III are true. Using only those statements, determine the truth values of the three propositions JB(c, g), JB(b, w), and JB(g, b). You may use a truth table or a deductive sequence proof. You may find it useful to abbreviate the three propositions involved as p, q, and r.

  • (b, 20) Asssuming that Statements I-V are all true, and assuming that the relation JB is antireflexive, antisymmetric, and transitive, prove that Statement VI is true. You may use either English or symbols, but make it clear each time you use a quantifier proof rule.

• Question 3 (30): The following are fifteen true/false questions, with no explanation needed or wanted, no partial credit for wrong answers, and no penalty for guessing. Some use the sets, relations, and functions defined above, but you should assume the truth of Statements I-VI only if explicitly told to. Two points for each correct answer.
  • (a) If Statements I-VI are all true, and the relation JB has the specified properties, then there must be at least six dogs in the set S.

  • (b) If Statements I-VI are all true, and the relation JB has the specified properties, then the negation of the relation JB is not a partial order.

  • (c) If Statements I-VI are all true, and the relation JB has the specified properties, then the relation R, defined so that R(x, y) is true if and only if JB(x, y) and JB (y, x) are both false, is an equivalence relation.

  • (d) Ignoring all the other statements, Statement II is true in exactly five of the eight possible settings of its three atomic variables.

  • (e) The following is a tautology: (p → (q → r)) ↔ ((p → q) → r)

  • (f) (A ∩ B) Δ C = (A ∩ C) Δ (B ∩ C) is a set identity.

  • (g) For any predicates P and Q we have [∀x: P(x) ∨ Q(x)] ↔ [(∀y: P(y)) ∨ (∀z: Q(z))].

  • (h) If a binary relation R on some set is symmetric, then its complement R_c(x, y) = ¬R(x, y) may fail to be symmetric.

  • (i) For any functions f and g, if g ∘ f is bijective and f is onto, then g is one-to-one.

  • (j) If X is any set and P is a predicate over X, then from ∀x: P(x) we can conclude by Specification that there exists some value v ∈ X for which P(v) holds.

  • (k) If X is any language that contains no strings of odd length, then the language XX (the concatenation product of X with itself) also contains no strings of odd length.

  • (l) If u and v are nonempty strings, vu is a suffix of w, and u is not a suffix of v, then v is not a suffix of w.

  • (m) If n is a positive natural, the relation R(x, y) defined as x = y ∨ xy ≡ 1 (mod n) is not an equivalence relation.

  • (n) The numbers 97, 115, and 119 can each be written in the form pⁱq^j where p and q are prime numbers and i and j are positive naturals.

  • (o) Let x and y be two naturals such that neither is divisible by 119. Then it is possible that xy is divisible by 119.

• Question 4 (30): Here are some straightforward number theory questions.
  • (a, 10) Prove that the three naturals 97, 115, and 119 are pairwise relatively prime, by using the Euclidean Algorithm on each of the three pairs.

  • (b, 10) Find an inverse of 97, modulo 119, and an inverse of 119, modulo 97. Show your calculations. Be sure to make clear which is which.

  • (c, 10XC) Mr. Lear, an elderly man with three daughters, is making arrangements for his retirement. His bank account is accessed by a four-digit PIN number, which we may think of as a natural number x that is less than 10000.

    He gives each of his daughters partial information about x, so that none of them can determine x on her own. He tells Cordelia the remainder x % 97 from dividing x by 97. He tells Goneril the number x % 115, and Regan the number x % 119.

    Explain why any two of the daughters, by combining their information, can determine x. You may quote the Chinese Remainder Theorem without proof if you do so accurately.

Last modified 18 October 2018