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: 20 points
Q2: 15 points
Q3: 15 points
Q4: 15 points
Q5: 10 points
Q6: 25+10 points
Total: 100+10 points
```

• Question 1 (20): Translate each statement as indicated, using the set of dogs D that includes the three distinct dogs Cardie, Duncan, and Nala (denoted by c, d, and n), perhaps with others, the set of conditions C, including the three distinct conditions Abandoment, Spanish Inquisition, and Thunder (denoted by a, s, and t), perhaps with others, and the three predicates F(x, y) meaning "dog x fears condition y", E(x, y) meaning "dog x expects condition y", and T(x) meaning "dog x is a terrier".

• (a, 2) (to English) (Statement I) (F(c, a) → E(c, a)) ∧ (¬F(c, a) → ¬E(c, a))
• (b, 3) (to symbols) (Statement II) If Nala expects thunder, then Duncan expects the Spanish Inquisition.
• (c, 2) (to English) (Statement III) ¬F(n, t) ∨ F(c, a)
• (d, 3) (to symbols) (Statement IV) Nala fears anything that she does not expect.
• (e, 2) (to English) (Statement V) ∀x: ¬E(x, s)
• (f, 3) (to symbols) (Statement VI) Terriers fear nothing.
• (g, 2) (to English) (Statement VII) ∀x:∀y:∀z: (E(x, y) ∧ E(x, z)) → (y = z)
• (h, 3) (to symbols) (Statement VIII) There is no dog that does not expect any condition.

• Question 2 (15): This question uses the definitions, premises, and predicates from Question 1. Do not draw any inferences from the English meaning of the predicates.

Using Statements I, II, III, IV, and V only, determine the value of the five propositions E(c, a), E(d, s), E(n, t), F(c, a), and F(n, t). (Hint: You will want to use the Rule of Specification to get information about these propositions from Statements IV and V.)

• 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 using Statements IV, V, VI, VII, and VIII only that Nala is not a terrier. Make your use of the quantifier proof rules clear.

• Question 4 (15): This question uses Statements IV, V, VI, VII, and VIII from Question 1. Do not make inferences from the English meanings of the words -- just use those statements.

• (a, 5) Explain why the relation E must be a function if all those statements are true.
• (b, 5) Can we tell from those statements alone whether E is a one-to-one function (an injection)? Explain why or why not. Remember that there may be other dogs or conditions along with the named ones.
• (c, 5) Can we tell from those statements along whether E is an onto function (a surjection)? Explain why or why not. Remember that there may be other dogs or conditions along with the named ones.

• Question 5 (10): Again we use Statements IV, V, VI, VII, and VIII from Question 1. Do not make inferences from the English meanings of the words. Define a binary relation R on the set D, by the rule R(x, y) ↔ (&exists; z: F(x, z) ∧ F(y, z)).

• (a, 2) Is R reflexive? Explain your answer, which might be "yes", "no", or "we can't tell".
• (b, 2) Is R symmetric? Explain your answer, which might be "yes", "no", or "we can't tell".
• (c, 2) Is R antisymmetric? Explain your answer, which might be "yes", "no", or "we can't tell".
• (d, 2) Is R transitive? Explain your answer, which might be "yes", "no", or "we can't tell".
• (e, 2) Is R either an equivalence relation or a partial order? Explain your answer, which might be "yes", "no", or "we can't tell".

• Question 6 (25+10): These five questions all involve number theory, mostly with the natural numbers 49 and 72.

• (a, 5) Explain, without using the Euclidean Algorithm, how you know that 49 and 72 are relatively prime.
• (b, 5) Show all the steps of the Euclidean Algorithm on inputs 49 and 72.
• (c, 10) Find a multiplicative inverse of 49 (modulo 72), and a multiplicative inverse of 72 (modulo 49).
• (d, 5) Suppose that I know, for some integer x, that x ≡ 42 (mod 49) and that x ≡ 42 (mod 72). State clearly what the Simple Form of the Chinese Remainder Theorem tells us about x.
• (e, 10XC) Suppose now that we know x ≡ 1 (mod 2), x ≡ 2 (mod 3), and x ≡ 2 (mod 5). What are the possible values of x in the range from 200 through 300? (Note: You need the statement of the Full Form of the Chinese Remainder Theorem to solve this, but not the proof in the book, since brute-force search is practical on such small numbers.)