# First Midterm Exam

### Directions:

• Answer the problems on the exam pages.
• There are four problems for 100 total points plus 10 extra credit. Actual scale is A = 85, C = 55.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.
• In case of a numerical answer, an arithmetic expression like "32× 26 + 13 × 41" need not be reduced to a single integer.

```  Q1: 20 points
Q2: 15 points
Q3: 35 points
Q4: 30+10 points
Total: 100+10 points
```

• Several questions concern a group of dogs ( Ace, Biscuit, Cardie, and Duncan, denoted by the constants a, b, c, and d respectively) and a scientist's observations of their behavior. If u and v are dogs, the predicate X(u, v) means "u is dominant over v" or equivalently "v is dominated by u". Do not assume anything about this predicate unless you are explicitly given that assumption as a premise to a problem. In fact, some questions do not assume that you are working with those particular four dogs, but with any set of dogs and any binary relation X on them.

• Question 1 (20): Translate the following seven statements as indicated:

• (a) Statement I (to English): [X(c, d) ∨ X(d, c)] → [X(a, b) ∧ ¬X(a, b)]

• (b) Statement II (to symbols): If Ace is not dominant over Cardie, then either Cardie is dominant over Duncan, or Duncan is dominant over Ace, but not both.

• (c) Statement III (to English): [¬X(b, d) → X(c, a)] ∧ [¬X(c, a) → ¬X(b, d)]

• (d) Statement IV (to symbols): For some dog, given any dog other than it, one of those two dogs is dominant over the other.

• (e) Statement V (to English): ∀u:∀v:(¬X(u, v)) ∨ (¬X(v, u))

• (f) Statement VI (to symbols): Given any dog, there are a second and a third dog (neither equal to the first dog nor equal to each other), such that neither the first dog nor the second is dominant over the other, and the third dog is dominant over both the first and the second.

• (g) Statement VII (to symbols): There is a dog who neither dominates nor is dominated by any other dog.

• Question 2 (15): Assuming that Statements I, II, III, V, and VII are true, determine all sixteen values of X(u, v) where u and v are taken from the set {Ace, Biscuit, Cardie, Duncan). (You may find it useful to do Question 3(a) first.)

• Question 3 (35): Here are some more questions about the relation X. Here you are not necessarily assuming the truth of all of the statements you used in Question 2 -- use only the statements you are given in each part.

• (a, 5) A relation R is defined to be antireflexive if ∀u: ¬R(u, u). Prove that if X is any relation that makes Statement V true, then X is antireflexive.

• (b, 10) A dominance chain is a sequence of dogs u1, u2,..., uk (where k is any positive natural) such that X(ui, ui+1) is true for any i in the range from 1 through k-1. Let Y(u, v) be defined to mean "there exists a dominance chain with u1 = u and uk = v". Explain in English why Y must be a transitive relation for any X. Is Y guaranteed to be antisymmetric if X is? Justify your answer.

• (c, 10) Define the relation Z so that Z(u, v) means "(u = v) ∨ Y(u, v)". Prove carefully that if Y is antisymmetric, then Z is a partial order. Draw the Hasse diagram for the Z created from the X given in Question 2.

• (d, 10) Prove that Statements IV, V, and VI cannot all hold at the same time for the same relation X, by deriving a contradiction from the three of them.

• Question 4 (30+10): My dog Duncan needs a new case of canned food every 24 days and a new bag of dry food every 35 days.

• (a,5) Using the Euclidean Algorithm, show that 24 and 35 are relatively prime.

• (b,10) Find integers x and y such that 24x + 35y = 1.

• (c,5) Quoting results from the course, determine how often I run out of canned food and dry food on the same day. (An arithmetic expression is enough -- you don't have to evaluate it.)

• (d,10) If I started a new case of canned food 6 days ago, and started a new bag of dry food 13 days ago, how many days will it be until I next run out of both kinds of food on the same day?

• (e,10XC) Let S be the set of numbers {0, 1, 2,..., 34} and let the function f from S to S be defined by f(n) = (24n + 9) % 35, where "%" is the Java modular division operator. Explain carefully why we know that f is a bijection (both one-to-one and onto) and determine the inverse function for f.