- Answer the problems on the exam pages.
- There are four problems, each with multiple parts, for 100 total points plus 5 extra credit. Actual scale A = 93, C = 63.
- Some useful definitions precede the questions below.
- No books, notes, calculators, or collaboration.
- In case of a numerical answer, an arithmetic expression like
"2
^{17}- 4" need not be reduced to a single integer.

Q1: 20 points Q2: 30 points Q3: 30 points Q4: 20+5 points Total: 100+5 points

Here are definitions of sets, predicates, and statements used on this exam.

Remember that the score of any quantifier is always to the end of the statement it is in.

Question 2 deals with the following scenario. Every day, Duncan monitors all visitors to the house and evaluates which ones (in his judgment) pose a threat. One day there were exactly five visitors, arriving at five distinct times. In alphabetical order, they were an Amazon driver (ad), the house cleaner (hc), the mail person (mp). a political canvasser (pc), and a UPS driver (ud).

Let V be the set {ad, hc, mp, pc, ud}. Let PT and DA be two unary relations on V, such that PT(x) means "visitor x posed a threat" and DA(x) means "visitor x was driven away by Duncan". Let AB be a binary relation on V, such that AB(x, y) means "visitor x arrived before visitor y", or equivalently "visitor y arrived after visitor x". We assume that AB is a strict total order, so that it is antireflexive, antisymmetric, transitive, and total.

Let **N** be the set of natural numbers {0, 1, 2, 3,...}.

If a, b, and m are naturals, with m > 0, the notation "a ≡ b (mod m)" means "a is congruent to b, modulo m".

The operator "%" on naturals, as in Java, refers to integer division, so that "x % y" is the remainder on dividing x by y.

**Question 1 (20):**Translate each statement as indicated, using the set of visitors V = {ad, hc, mp, pc, ud} defined above and the predicates PT(x), DA(x), and AB(x, y) defined above to mean "visitor x posed a threat", "visitor x was driven away by Duncan", and "visitor x arrived before visitor y" respectively. All variables should be of type "visitor". The predicate AB is assumed to be a strict total order (antireflexive, antisymmetric, transitive, and total).- (a, 3) (to English) (Statement I)
(PT(pc) ↔ AB(hc, pc)) ∧ (PT(hc) → (PT(pc) ∧ AB(hc, pc)))
- (b, 3) (to symbols) (Statement II) If both the political canvasser and the house cleaner posed a threat, then the house cleaner arrived before the political canvasser.
- (c, 3) (to English) (Statement III) (PT(pc) ∨ PT(hc) ∨ AB(hc, pc)) ∧ ¬(AB(hc, pc) ∧ PT(hc) ∧ PT(pc))
- (d, 4) (to symbols) (Statement IV) Every visitor who was not driven away by Duncan did not arrive after the house cleaner.
- (e, 4) (to English) (Statement V) ∃y:∀z:DA(y)∧ (AB(z, hc) ↔ (y=z))
- (f, 3) (to symbols) (Statement VI) Every visitor who posed a threat was driven away by Duncan.

- (a, 3) (to English) (Statement I)
(PT(pc) ↔ AB(hc, pc)) ∧ (PT(hc) → (PT(pc) ∧ AB(hc, pc)))
**Question 2 (30):**These questions use the definitions, predicates, and premises above.- (a, 10)
Given
*only*that Statements I, II, and III are true, determine the truth values of the three propositions q = PT(hc), r = PT(pc), and s = AB(hc, pc). You may use a truth table or a deductive sequence proof. - (b, 20) Asssuming that Statements I-V are all true, and assuming that the relation AB is a strict total order (antireflexive, antisymmetric, transitive, and total) prove that Statement VI is also true. You may use either English or symbols, but make it clear each time you use a quantifier proof rule.

- (a, 10)
Given
**Question 3 (30):**The following are fifteen true/false questions, with no explanation needed or wanted, no partial credit for wrong answers, and no penalty for guessing. Some use the sets, relations, and functions defined above, but you should assume the truth of Statements I-VI only if explicitly told to. Two points for each correct answer.- (a) If Statements I-VI are all true, and the relation AB has the specified properties, we do not know whether the mail person posed a threat.
- (b) There exists a symmetric binary relation on some non-empty set that is not reflexive.
- (c) If n is any natural with n > 1, then n and n+1 are relatively prime.
- (d) Let the set S be {2, 4, 8, 26, 32} and define a partial order by the division relation D(a, b), which means that a divides b with no remainder. Then when we construct the Hasse diagram, there are fewer than five edges.
- (e) If |A| = 4 and |B| = 5, then there is not any surjective (onto) function from A to B.
- (f) Let A and B be any two disjoint nonempty sets (so that A ∩ B = ∅). Let U be any partial order on A and let V be any partial order on B. Then U ∪ V (the set of all ordered pairs that are in either U or V) is a partial order on A ∪ B.
- (g) Every total order is also a partial order.
- (h) If R is an equivalence relation on a nonempty finite set A, then the equivalence classes of R must each have the same number of elementss.
- (i) The function f:
**N**→**R**, defined by f(n) = √n, is an injection (1-1 function) but not a surjection (onto function). - (j) The negation of ∀x:DA(x) is ¬∀x:¬DA(x).
- (k) If f(x) = 2x
^{2}+ 1 and g(x) = 4√x, then (f ○ g)(x) = 4√2x^{2}+ 1. - (l) Every one-to-one function has an inverse function.
- (m) If we know that p is true, and r is any proposition at all, then (p ∨ r) must also be true.
- (n) If we know that p → q is true, and r is any proposition at all, then (p ∨ r) → q must also be true.
- (o) Let R be a relation defined on the set
**Z**so that R(a, b) if and only if a ≠ b. Then R is symmetric and transitive, but not reflexive.

**Question 4 (20+5):**Here are some straightforward number theory questions. Let X be the set of naturals {0, 1, 2,..., 99}. Define the function g from X to X by the rule g(x) = (77x)%100, where "%" denotes the Java remainder operator.- (a, 5) Give prime factorizations of the naturals 77 and 100, and explain why these factorizations prove that these numbers are relatively prime to one another.
- (b, 5) Compute integers y and z such that 77y + 100z = 1. You could conceivably do this by brute force, but the Inverse Algorithm from lecture is far easier and more reliable.
- (c, 5) Prove that the function g from X to X, defined above, is a bijection.
- (d, 5) Find a function from X to X that is an inverse of g.
- (e, 5XC) For any numbers a and B in X, define the function
f
_{a, b}from X to X by the rule f_{a, b}(x) = (ax+b)%100. For what values of a and b is f_{a, b}a bijection? Prove your answer.

Last modified 2 March 2020