# First Midterm Exam

### Directions:

• Answer the problems on the exam pages.
• There are six problems, some with multiple parts, for 100 total points plus 10 extra credit. Estimated scale A = 93, C = 69, but may be adjusted.
• 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: 15 points
Q2: 15 points
Q3: 15 points
Q4: 15 points
Q5: 10 points
Q6: 30+10 points
Total: 100+10 points
```

Correction in orange made 25 Feb 2014.

• Question 1 (15): Translate each statement as indicated, using the set of dogs {Cardie, Duncan, Nala} (denoted by c, d, and n, with no two equal to one another), and the two predicates R(x, y) meaning "dog x is equal to or faster than dog y" and G(x, y) meaning "dog x's favorite dog is dog y".

• (a, 3) (to English) (Statement I) ∀x: R(n, x)
• (b, 3) (to symbols) (Statement II) There is a dog that Nala is equal to or faster than, and that dog is Nala's favorite dog.
• (c, 3) (to symbols) (Statement III) Given any two dogs, if the second dog is not Cardie, then the first dog is equal to or faster than the second dog if and only if the first dog's favorite dog is not the second dog.
• (d, 3) (to English) (Statement IV) ¬∃u:∃v:∃w:G(u, v) ∧ G(u, w) ∧ (v ≠ w)
• (e, 3) (to symbols) (Statement V) It is not the case that if Cardie's favorite dog is Nala, then Duncan is equal to or faster than Nala.

• Question 2 (15): This question uses the definitions premises, and predicates from Question 1.

Prove from the premises that Nala's favorite dog is not Duncan. Use only the premises, not any inferences from the English meaning of the predicates. (Hint: This can be done by using Specification on Statements I and III, then just propositional logic.)

• Question 3 (15): This question also uses the definitions, predicates, and premises from Question 1. Again, do not make any inferences from the English meaning of the predicates.

Prove from the premises that the relation G is a function from S to S.

• Question 4 (15): This question uses the definitions from Question 1, but not the premises. For this question only, you may use the normal English meaning of "equal to or faster than". In addition, you may assume that given any two different dogs, one is faster than the other.

• Question 5 (10): This question uses the definitions, predicates, and premises of Question 1. Do not use any inferences from the English meaning of the predicates. Assume the result of Question 3, so that G is a function. Given what you can determine from the premises:

• (a, 5) Is G a one-to-one function (an injection)? Explain your answer.
• (b, 5) Is G an onto function (a surjection)? Explain your answer.

• Question 6 (30+10): These four questions involve the natural numbers 24 and 35, which are unfortunately the same pair of numbers I chose for the Spring 2012 exam.

• (a, 10) Use the Euclidean Algorithm to show that 24 and 35 are relatively prime.
• (b, 10) Find integers x and y so such that 24x + 35y = 1. Once you have found those integers, give a multiplicative inverse of 24 (modulo 35) and a multiplicative inverse of 35 (modulo 24).
• (c, 10) Suppose that I know, for some integer x, that x ≡ 2 (mod 35) and that x ≡ 3 (mod 24). State clearly what the Simple Form of the Chinese Remainder Theorem tells us about x.
• (d, 10XC) Find an integer x such that x ≡ 2 (mod 35) and that x ≡ 3 (mod 24). (Hint: Your answer to part (b) should be useful, though if you have time you might be able to find the number by trial and error.