CIS 375 Intro to Discrete Mathematics Exam 3 (Section M004: Blue) 6 December Points Possible
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1 Name: CIS 375 Intro to Discrete Mathematics Exam 3 (Section M004: Blue) 6 December 2016 Question Points Possible Points Received 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.
2 1. (12 points) Consider the poset (A, ) given by the following Hasse diagram: D W P B C G F T A U X (a) List all of the maximal elements of the poset. (b) List all of the minimal elements of the poset. (c) List all of the elements that are comparable with F. (d) List all of the elements that are not comparable with A.
3 2. (14 points) (a) Let R g be the following relation: R g = {(2, a), (4, b), (6, c), (8, d)}. i. The relation R g is a function. Give the image of R g. ii. Give a pair that, if added to R g, would yield a relation that is not a function. iii. Give a pair that, if added to R g, would yield a function that is not 1-1. iv. Give sets A and B such that R g is a function from A to B that is not onto. (b) Give a function h : N Z that is 1-1 and not onto. (c) Give a function f : Z Z that is onto and not 1-1.
4 3. (14 points) (a) How many elements (i.e., pairs) are in the largest partial order on the set {1, 2, 3, 4, 5, 6}? (b) How many maximal elements does the poset (2 {2,4,6,8}, ) have? (c) How many elements (i.e., pairs) are in the smallest partial order on the set {1, 2, 3, 4, 5}? (d) How many elements (i.e., pairs) are in the smallest partial order R such that {(3, 4), (5, 2), (4, 7)} R? (e) Give a relation on the set {s, t, w, q} that is a partial order and not an equivalence relation. (f) Give a Hasse diagram for a five-element poset that has exactly two minimal elements and four maximal elements.
5 4. (12 points) Consider the set A of strings, which is defined recursively as follows: The string 302 is in A. The string 12 is in A. If u is a string in A, then the string 00uu is also in A. If s and w are strings in A, then the string s11w2 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.
6 5. (16 points) Use mathematical induction to prove the following claim: For every integer m 0, m (3i + 1) = i=1 m(3m + 5). 2
7 6. (16 points) Use mathematical induction to prove the following claim: For all integers q 1, 6 q q 1 is divisible by 43.
8 7. (16 points) Consider the set T Z Z defined recursively as follows: The pair (0, 2) is in T. The pair (1, 6) is in T. If the pairs (m, k) and (m + 1, j) are both in T, then the pair (m + 2, 6j 8k) is in T. Use structural induction to prove the following claim: For all (a, b) T, b = 2 2a + 2 a.
9 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.
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