- Answer the problems on the exam pages.
- There are three problems on pages 2-7, for 100 total points plus 10 extra credit. 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: 30 points Q2: 50 points plus 10 extra credit Q3: 20 points Total: 100 points plus 10 extra credit

**Question 1 (30):**Let a be the proposition "he is asleep", c be the proposition "he has his coat", l be the proposition "he has the leash" and w be the proposition "it is time for a walk".- (a,10) Translate the following four statements as indicated:
- (to English) (I) (a ∧ c) ⊕ l
- (to symbols) (II) If he has the leash, then he also has his coat.
- (to English) (III) l ∨ (c → ¬a)
- (to symbols) (IV) It is time for a walk if and only if it is not the case that either he is asleep, he does not have the leash, or he does not have his coat.

- (b,20) Prove that given the statements I, II, III, and IV above, it is
time for a walk. One good way to do this is to construct a truth table.
Another is to use a deductive sequence proof -- a good way to do this is to
use Proof by Cases with intermediate proposition l. In a deductive
sequence proof, remember that you may use
*valid*rules even if you don't remember their names.

- (a,10) Translate the following four statements as indicated:
**Question 2 (50+10):**This question involves a set of dogs D and the following predicates: L(x) means "dog x is a Labrador", S(x) means "dog x likes to swim", and B(x) means "dog x is black".- (a,10) Translate the following three statements as indicated:
- (to symbols) (V) Ebony is a black Labrador.
- (to English) (VI) ∃y: S(y) ∧ ¬B(y)
- (to symbols) (VII) Only Labradors like to swim.

- (b,10) The following statement VIII defines the binary predicate
E(x,y) on D:
¬E(x,y) ↔ (B(x) ⊕ B(y))

Prove that E is an equivalence relation. (Use this definition only, not the statements from (a).)

- (c,20) Using statements V, VI, VII, and VIII from parts (a) and (b),
prove the statement:
∀u:∃v: L(v) ∧ E(u,v)

(For full credit you must use the predicate calculus proof rules -- there will be partial credit for informal arguments.)

- (d,10) Let C be the set {b,n} and define a function f from D to C by the rules (f(x) = b) ↔ B(x) and (f(x) = n) ↔ ¬B(x). Can you determine from statements V, VI, and VII in part (a) whether the function f is onto (a surjection)? Justify your answer, making it clear that you understand the relevant definitions.
- (e,10 XC) Can you determine from statements V, VI, and VII in part (a) whether the function f is one-to-one (an injection)? Justify your answer, making it clear that you understand the relevant definitions.

- (a,10) Translate the following three statements as indicated:
**Question 3 (20):**Let**N**= {0,1,2,3,...} be the set of all naturals. Let R be the binary relation on**N**defined by:R(a,b) ↔ [(a ≤ b) ∧ (∃c: b - a = 2c)]

That is, R(a,b) means that b - a is an even natural.

Here are your questions:

- (a,10) Remember that the definition for R to be
**antisymmetric**is that ∀a:∀b:[R(a,b) ∧ R(b,a)] → (a = b). Argue carefully that R is antisymmetric. You may use standard facts of arithmetic without proof. - (b,10) State the other two properties necessary to show that R is a partial order. Argue carefully that R has these two properties.

- (a,10) Remember that the definition for R to be

Last modified 23 February 2005