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

Q1: 15 points Q2: 15 points Q3: 15 points Q4: 20 points Q5: 20 points Q6: 15 points Total: 100 points

**Note for Questions 1-4:**These questions deal with a set of dogs D = {Chinook, Ebony, Hadley}, a set of breeds B = {Husky, Labrador, Poodle}, and a binary predicate M defined so that M(d,b) means "dog d is a member of breed b''.**Question 1 (15):**Translate each of the following English statements into symbolic notation:- (a,5) If Hadley is a Labrador, then Chinook is not a Husky.
- (b,5) Hadley is either a Labrador or a Poodle, but not both, and if Chinook is either a Husky or a Poodle, but not both, then Hadley is not a Poodle.
- (c,5) Every dog is a member of some breed.

**Question 2 (15):**Translate each of the following symbolic statements into English. (The type of variables b, b_{1}, and b_{2}is "breed", and the type of d, d_{1}, and d_{2}is "dog".)- (d,5) (M (Chinook, Husky) → M (Hadley, Labrador))∧ (M(Hadley, Labrador) → M(Chinook, Husky)).
- (e,5)
∀ d
_{1}:∀ d_{2}: ∀ b: [M(d_{1},b)∧ M(d_{2},b)∧ (d_{1}≠d_{2})]→ (b = Labrador) - (f,5)
∀ b
_{1}:¬∃ d:∃ b_{2}: M(d,b_{2})∧ (b_{1}≠b_{2})∧ M(d,b_{1}).

**Question 3 (15):**Assume that statements (a) through (f) from Questions 1 and 2 are true, and further assume that Ebony is a Labrador. You may find it useful to solve parts of Question 4 first, or otherwise determine the breeds of the various dogs.- (g,5) Prove, using the definition of a function, that the relation M is a function from the given set of dogs to the given set of breeds.
- (h,5) Is M a one-to-one function? Prove your answer using the given assumptions.
- (i,5) Is M an onto function? Prove your answer using the given assumptions.

**Question 4 (20):**This question deals with some of the propositions obtained by substituting specific values into M.- (j,5) Using a truth table, determine the truth value of M (Chinook, Husky) and M (Hadley, Labrador) from statements (a) and (d) only.
- (k,10) Using
*only*information about the propositions M (Chinook, Husky), M (Chinook, Poodle), M (Hadley, Labrador), and M (Hadley, Poodle) from statements (a), (b), (d), and (e), determine the value of these four propositions and prove your answer from the statements using propositional proof rules. If you are sure that a rule is valid but can't remember its name, just state it and use it. - (l,5) Verify that your conclusion is consistent with statement (f).

**Question 5 (20):**The binary relation Pre on strings is defined so that Pre(u,v) means ∃w:uw=v. If this is true we say that u is a*prefix*of v. Prove carefully that Pre is a partial order on the set of all strings over any alphabet. State the three properties of a partial order, and use the definition of Pre and quantifier proof rules to show that each of the three properties holds. (Hint: You may assume that every string has a*length*, and that the length of xy is the length of x plus the length of y. You may also assume well-known properties of the naturals.)**Question 6 (15):**Recall that we may define two naturals a and b to be*congruent*modulo a natural r if there exists a natural q such that either a=qr+b or b=qr+a. We write this "a≡b (mod r)". Recall also that a natural c*divides*a natural d (written "D(c,d)") if and only if ∃ e:ce=d. Prove carefully, using exactly these definitions, that: ∀a:∀b:∀c:∀d:[D(a,b)∧ (c ≡ d (mod b))]→ (c ≡ d (mod a))

Last modified 29 September 2004