COMP9020 Assignment 2 2019 Term 3Due: Sunday, 3rd November, 23:59Submission is through WebCMS/give and should be a single pdf file, maximum size 2Mb. Prose shouldbe typed, not handwritten. Use of LATEX is encouraged, but not required.Discussion of assignment material with others is permitted, but the work submitted must be your own inline with the University’s plagiarism policy.Problem 1 (20 marks)Recall the relation composition operator ; defined as:R1; R2 = {(a, c) : there is a b with (a, b) ∈ R1 and (b, c) ∈ R2}For any set S, and any binary relations R1, R2, R3 ⊆ S × S, prove or give a counterexample to disprove thefollowing:(a) (R1; R2); R3 = R1;(R2; R3) (4 marks)(b) I; R1 = R1; I = R1 where I = {(x, x) : x ∈ S} (4 marks)(c) (R1; R2)← = R←(4 marks)(d) (R1 ∪ R2); R3 = (R1; R3) ∪ (R2; R3) (4 marks)(e) R1;(R2 ∩ R3) = (R1; R2) ∩ (R1; R3) (4 marks)Problem 2 (30 marks)Let R ⊆ S × S be any binary relation on a set S. Consider the sequence of relations as follows:(a) Prove that if there is an i such that Rfor all j ≥ i. (4 marks)(b) Prove that if there is an i such that Rfor all k ≥ 0. (4 marks)(c) Let P(n) be the proposition that for all m ∈ N: Rn+m. Prove that P(n) holds for all n ∈ N.(8 marks)(d) If |S| = k, explain why Rk = Rk+1. (Hint: Show that if (a, b) ∈ Rk+1then (a, b) ∈ Rifor some i (4 marks)(e) If |S| = k, show that Rkis transitive. (4 marks)(f) If |S| = k, show that (R ∪ R←)kis an equivalence relation. (6 marks)1Problem 3 (26 marks)A binary tree is a data structure where each node is linked to at most two successor nodes:If we allow empty binary trees (trees with no nodes), then we can simplify the description by saying anode has exactly two children which are binary trees.(a) Give a recursive definition of the binary tree data structure. Hint: review the recursive definition of aLinked List (6 marks)A leaf in a binary tree is a node that has no successors (i.e. it has two empty trees as children). A fullyinternalnode in a binary tree is a node that has two successors. The example above has 3 leaves and 2fully-internal nodes.(b) Based on your recursive definition above, define the function count(T) that counts the number of nodesin a binary tree T. (4 marks)(c) Based on your recursive definition above, define the function leaves(T) that counts the number ofleaves in a binary tree T. (4 marks)(d) Based on your recursive definition above, define the function internal(T) that counts the number offully-internal nodes in a binary tree T. Hint: it is acceptable to define an empty tree as having −1 fullCOMP9020代做、代写WebCMS、R设计代做、代写R程yinternalnodes. (4marks)(e) If T is a binary tree, let P(T) be the proposition that leaves(T) = 1 + internal(T). Prove that P(T) holdsfor all binary trees T. (8 marks)Problem 4 (24 marks)Four wifi networks, Alpha, Bravo, Charlie and Delta, all exist within close proximity to one another asshown below.Alpha Bravo Charlie DeltaNetworks connected with an edge in the diagram above can interfere with each other. To avoid interferencenetworks can operate on one of two channels, hi and lo. Networks operating on different channels will notinterfere; and neither will networks that are not connected with an edge.Our goal is to determine (algorithmically) whether there is an assignment of channels to networks sothat there is no interference. To do this we will transform the problem into a problem of determining if apropositional formula can be satisfied.2(a) Carefully defining the propositional variables you are using, (4 marks)write propositional formulas for each of the following requirements:(i) ϕ1: Alpha uses channel hi or channel lo; and so does Bravo, Charlie and Delta. (4 marks)(ii) ϕ2: Alpha does not use both channel hi and lo; and the same for Bravo, Charlie and Delta.(4 marks)(iii) ϕ3: Alpha and Bravo do not use the same channel; and the same applies for all other pairs ofnetworks connected with an edge. (4 marks)(b) (i) Show that ϕ1 ∧ ϕ2 ∧ ϕ3 is satisfiable; so the requirements can all be met. Note that it is sufficientto give a satisfying truth assignment, you do not have to list all possible combinations. (4 marks)(ii) Based on your answer to the previous question, which channels should each network use in orderto avoid interference? (4 marks)3Advice on how to do the assignmentAll submitted work must be done individually without consulting someone else’s solutions in accordancewith the University’s “Academic Dishonesty and Plagiarism” policies.• Assignments are to be submitted via WebCMS (or give) as a single pdf file.• Be careful with giving multiple or alternative answers. If you give multiple answers, then we willgive you marks only for your worst answer, as this indicates how well you understood the question.• Some of the questions are very easy (with the help of the lecture notes or book). You can use thematerial presented in the lecture or book (without proving it). You do not need to write more thannecessary (see comment above).• When giving answers to questions, we always would like you to prove/explain/motivate your answers.• If you use further resources (books, scientific papers, the internet,...) to formulate your answers, thenadd references to your sources.4转自：http://www.3daixie.com/contents/11/3444.html