Math 3000 Section 003 Intro to Abstract Math Final Exam

Transcription

1 Math 3000 Section 003 Intro to Abstract Math Final Exam Department of Mathematical and Statistical Sciences University of Colorado Denver, Spring 2012 Name: Problem 1a-j 2 3a-b 4a-b 5a-c 6a-c 7a-b 8a-j Total Minutes Points Score Read all problems carefully and write your solutions concisely in complete sentences. For each proof, state its type, all assumptions and basic definitions, and give a concluding statement. 1. For each of the following items (a)-(j), select one of the given three concepts or results (check the corresponding box) and define, explain, or state it in one or two complete, grammatically and mathematically correct sentences (and with correct mathematical notation). [2 pts each] (a) Sets: Cartesian product of two sets partition of a set proper subset of a set (b) Logic: necessary/sufficient condition contradiction/tautology De Morgan s laws (c) Proofs: proof by contradiction proof by contrapositive existence/uniqueness proof (d) Induction: anchor weak/strong induction proof by minimum counterexample

2 Math 3000 Section 003 Intro to Abstract Math Final Exam, UC Denver, Spring (e) Relations: relation (on a single set) symmetry and a(nti)symmetry partial order (f) Functions: function (from one set to another) range one-to-one correspondence (g) Cardinalities: infinite/uncountable set denumerable set sets of same cardinality (h) Equivalence: logical equivalence numerical equivalence equivalence relation/class (i) Divisibility: integers modulo n congruence modulo n same and opposite parity (j) Results: Goldbach s Conjecture Euclid s Lemma Schröder-Bernstein Theorem

3 Math 3000 Section 003 Intro to Abstract Math Final Exam, UC Denver, Spring Below are four fundamental results that we discovered in this class. Check one and prove it. Q is denumerable. The number 2 is irrational. R is uncountable. There are infinitely many primes. [10 points]

4 Math 3000 Section 003 Intro to Abstract Math Final Exam, UC Denver, Spring Consider the sequence F 1, F 2, F 3,... where F 1 = F 2 = 1, F 3 = 2, F 4 = 3, F 5 = 5, F 6 = 8, F 7 = 13, F 8 = 21, F 9 = 34, and F 10 = 55. These numbers are called the Fibonacci numbers. (a) Define the sequence of Fibonacci numbers by means of a recurrence relation. [2 points] (b) Prove that a Fibonacci number F n is even if and only if n is a multiple of 3. [8 points]

5 Math 3000 Section 003 Intro to Abstract Math Final Exam, UC Denver, Spring A grocery store sells rice only in bags of 3 and 5 pounds. (a) Find the minimum amount n 0 such that you can buy any n n 0 pounds of rice. [2 points] (b) Prove your finding (show that for every positive integer n n 0, there exist nonnegative integers a and b such that 3a + 5b = n, but not such that 3a + 5b = n 1). [8 points]

6 Math 3000 Section 003 Intro to Abstract Math Final Exam, UC Denver, Spring ( ) a b 5. Let M = Z c d 2 2 be a 2 2 integer matrix with determinant det(m) = ad bc, and n be a positive integer. A row or column of M is said to be divisible ( by n ) if n divides each of 4 10 that row s or column s entries (for example, the determinant of is 180, its first row is divisible by 2, its second row by 3, its first column by 4, and its second column by 5). (a) Prove that n det(m) if one row or column of M is divisible by n. [4 points] (b) Prove that n 2 det(m) if all rows and columns of M are divisible by n. [4 points] (c) Prove that the converse statements of (a) and (b) are not correct, in general. [2 points]

7 Math 3000 Section 003 Intro to Abstract Math Final Exam, UC Denver, Spring Let S and T be two (finite or infinite) sets. Prove or disprove each the following statements. (a) True False: S T = T S implies that S = T. [4 points] (b) True False: S T = T S implies that S = T. [4 points] (c) True False: S T = T S implies that S = T. [2 points]

8 Math 3000 Section 003 Intro to Abstract Math Final Exam, UC Denver, Spring Let a, b, c, and d be integers such that a + b 0 (mod 3) and c d (mod 3). (a) Prove that ac + bd 0 (mod 3). [5 points] (b) The above statement remains correct if 3 is replaced by any other positive integer n N: If a + b 0 (mod n) and c d (mod n), then ac + bd 0 (mod n). Evaluate whether your first proof still works or not. If not, give a new proof. [5 points]

9 Math 3000 Section 003 Intro to Abstract Math Final Exam, UC Denver, Spring Let R n be the set of real n-dimensional vectors for n 2. On Midterms 1 and 2, we have seen that the canonical, componentwise inequalities, and < are preorders and (strict) partial orders, but we still do not know whether there exist any total orders. In this exercise, you will answer that question affirmatively. Recall that a relation R on a set A is said to be connected (or complete) if for every two x and y in A, either x R y or y R x (or both); weakly connected if for every two distinct x y in A, either x R y or y R x (or both); a preorder if it is reflexive and transitive; a total preorder if it is reflexive, transitive, and connected; a partial order if it is reflexive, transitive, and antisymmetric; a strict partial order if it is irreflexive and transitive; a total order if it is asymmetric, transitive, and weakly connected. Let x = (x 1,..., x n ) and y = (y 1,..., y n ) be two vectors and define three relations on R n by x max y if max{x i : i = 1, 2,..., n} max{y i : i = 1, 2,..., n}; x < max y if max{x i : i = 1, 2,..., n} < max{y i : i = 1, 2,..., n}; x < lex y if x j < y j for some j and x i = y i for all i = 1, 2,..., j 1. These relations are called maximum-order (or max-order), strict maximum-order, and lexicographic order (or lex-order), respectively. [All following problems are 2 points each.] (a) The lex-order is also called alphabetical or dictionary order due to its analogy to sorting words in a dictionary. Explain this analogy using the words math and mahi. (b) Let n = 4, x = (1, e, 2, π), y = (1, e, π, 3). Which of the following statements are true? x max y x < max y x < lex y y max x y < max x y < lex x (c) Classify each of the above relations as total preorder, strict partial order, or total order. total preorder strict partial order total order max-order ( max ) strict max-order (< max ) lexicographic order (< lex ) (d) Are those relations that you classified as total preorders also preorders? Explain. (e) Are those relations that you classified as total preorders also partial orders? Explain.

10 Math 3000 Section 003 Intro to Abstract Math Final Exam, UC Denver, Spring (f) Are those relations that you classified as strict partial orders also partial orders? Explain. (g) Are those relations that you classified as total orders also (total) preorders? Explain. (h) If you classified a relation as strict partial order but not as total order, why (not)? (i) Prove that those relations that you classified as total preorders are connected. (j) Prove that those relations that you classified as total orders are weakly connected.