Name: CIS 375 Intro to Discrete Mathematics Exam 3 (Section M001: Green) 6 December 2016 Question Points Possible Points Received 1 12 2 14 3 14 4 12 5 16 6 16 7 16 Total 100 Instructions: 1. This exam is a closed-book, closed-notes exam. 2. Legibility counts! Make sure I can read (and find!) your answers. If you need more room for an answer than that given, use the back side of the pages. Be sure to leave a note indicating where the answer is. 3. This test should have 9 pages (including this cover sheet). Let me know now if your copy does not have the correct number of pages. 4. The last page of this exam lists some relevant definitions. Also recall these sets: Z: set of all integers R: set of all real numbers Z + = {z Z : z > 0} Z = {z Z : z < 0} N= {z Z : z 0} A reminder from algebra: For all real (or integer) numbers a, b, m, m a+b = m a m b.
1. (12 points) Consider the poset (A, ) given by the following Hasse diagram: T Q U J P W L N K (a) List all of the maximal elements of the poset. S (b) List all of the minimal elements of the poset. (c) List all of the elements that are comparable with J. (d) List all of the elements that are not comparable with N.
2. (14 points) (a) Consider the function h : {1, 2, 3, 4} {1, 2, 3, 4, 5, 6, 7, 8} defined by: { 2x + 1 x < 3 h(x) = 2x x 3 i. What is the image of h? ii. Is h an invertible function? iii. Every function can be viewed as a set of pairs: give h as a set of pairs. iv. Give a pair that, if added to h, would yield a function that is not 1-1. v. What is the domain of the function created in part (iv)? (b) Give a function f : N N Z that is onto and not 1-1. (c) Give a function g : Z Z + that is 1-1 and not onto.
3. (14 points) (a) Give a relation on the set {a, b, c, d, e} that is a partial order and not an equivalence relation. (b) How many elements (i.e., pairs) are in the smallest partial order R such that {(1, 6), (3, 5), (3, 6), (5, 7)} R? (c) How many maximal elements does the poset (2 {0,2,4,6,8}, ) have? (d) How many elements (i.e., pairs) are in the smallest partial order on the set {1, 2, 3, 4, 5}? (e) How many elements (i.e., pairs) are in the largest partial order on the set {1, 2, 3, 4, 5, 6}? (f) Give a Hasse diagram for a five-element poset that has exactly three maximal elements and four minimal elements.
4. (12 points) Consider the set A of strings, which is defined recursively as follows: The string 32 is in A. The string 012 is in A. If u is a string in A, then the string 3u3u is also in A. If s and w are strings in A, then the string 2sw2 is also in A. Thus, all elements of A are strings constructed from just four symbols: 0, 1, 2, and 3. (a) List five elements of A. (b) Suppose you were to use structural induction to prove that some property P holds of all elements of the set A. i. For the basis step, what would you need to show true? ii. For the inductive step, what conditional(s) would you need to show true? A Reminder for Inductive Proofs Follow the boilerplate for inductive proofs, including: Always label the basis, inductive step, and inductive hypothesis. Explicitly state when/where you use the inductive hypothesis. Include appropriate wrap-up statements. For this exam, you do not need to include the claim or the Proof: by induction preface.
5. (16 points) Use mathematical induction to prove the following claim: For every integer m 0, m (3i 2 + 5i) = m(m + 1)(m + 3). i=1
6. (16 points) Use mathematical induction to prove the following claim: For all integers q 1, 6 q 5q 1 is divisible by 25.
7. (16 points) Consider the set T Z Z defined recursively as follows: The pair (0, 7) is in T. The pair (1, 32) is in T. If the pairs (m, k) and (m + 1, j) are both in T, then the pair (m + 2, 9j 20k) is in T. Use structural induction to prove the following claim: For all (a, b) T, b = 4 5 a + 3 4 a.
A Collection of Some Relevant Definitions Subsets Let A and B be sets. A is a subset of B provided that the following condition holds: for all objects x, if x A then x B. Relations Let R X X be a relation. R is reflexive provided that the following condition holds: for all x X, (x, x) R. R is irreflexive provided that the following condition holds: for all x X, (x, x) R. R is symmetric provided that the following condition holds: for all x, y X, if (x, y) R then (y, x) R. R is antisymmetric provided that the following condition holds: for all x, y X, if (x, y) R and (y, x) R, then x = y. R is transitive provided that the following condition holds: for all x, y, z X, if (x, y) R and (y, z) R, then (x, z) R. R is an equivalence relation provided that R is reflexive, symmetric, and transitive. R is a partial order provided that R is reflexive, antisymmetric, and transitive. Functions Let f : X Y be a function. f is an injection (or 1-1) provided that the following condition holds: for all x 1, x 2 X, if f(x 1 ) = f(x 2 ) then x 1 = x 2. f is a surjection (or onto) provided that the following condition holds: for all y Y, there is a w X such that f(w) = y. f is a bijection provided that f is both 1-1 and onto.