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Let us define F≡ to be the set of equivalence classes of F under ≡. That is,F≡ := {[ϕ] :  ϕ ∈ F}.Part (c) above shows that the following operations are well-defined1 on F≡:[ϕ]∧ [ψ] defined to be [ϕ ∧ ψ]Boolean Algebra代写[ϕ]∨ [ψ] defined to be [ϕ ∨ ψ][ϕ]j defined to be[¬ϕ]Show that F≡together with the operations defined above forms a Boolean Algebra. Note: you  will have to give a suitable definition of a zero element and a one element in F≡. (12 marks)Boolean Algebra代写1well-defined means that the output is not dependent on different representations of the same input.Boolean Algebra代写Problem 2 (10 marks)This is the Petersen graph:(a)Givean argument to show that the Petersen graph does not contain a subdivision of K5. (5 marks)Boolean Algebra代写(b)Show that the Petersen graph contains a subdivisionof K3,3. (5 marks)Problem 3 (10 marks)Boolean Algebra代写Harry would like to take each of the following subjects: Defence against the Dark Arts; Potions; Herbology; Transfiguration; and Charms. Unfortunately some of the classes clash, meaning Harry cannot take them both. The list of clashes are:Defenceagainst the Dark Arts clashes with Potions and CharmsPotions also clashes with HerbologyHerbology also clashes with Transfiguration,andTransfiguration also clashes with Charms.Harry would like to know the maximum number of classes he can take.(a)Model this as a graph problem. Rememberto:(i)Clearlydefine the vertices and edges of your  (4 marks)(ii)State the associated graph problem that you need to  (2 marks)Boolean Algebra代写(b)Givethe solution to the graph problem corresponding to this scenario; and solve Harry’s problem.(4 marks)Problem 4 (22 marks) Boolean Algebra代写Recall from Assignment 2 the definition of a binary tree data structure: either an empty tree, or a node with two children that are trees.Let T(n) denote the number of binary trees with n nodes. For example T(3) = 5 because there are five binary trees with three nodes:Boolean Algebra代写(a)Using the recursive definition of a binary tree structure, or otherwise, derive a recurrence equation for T(n).A full binary tree is a non-empty binary tree where every node has either two non-empty children (i.e. is  a fully-internal node) or two empty children (i.e. is a leaf).(b)Using observations from Assignment 2, or otherwise, explain why a full binary tree must have an odd numberof  (4 marks)Boolean Algebra代写(c)Let B(n) denote the number of full binary trees with n Derive an expression for B(n), involving of T(nj ) where nj  ≤ n.  Hint:  Relate the internal nodes of a full binary treeto T(n). (6 marks)A well-formed formula is in Negated normal form if it consists of just , , and literals (i.e. propositional variables or negations of propositional variables).  That is, a formula that results after two steps of the process for transforming a formula into a logically equivalent one. For example, p ∨ (¬q ∧ ¬r) is in negated normal form; but p ∨ ¬(q ∨ r) is not.Boolean Algebra代写Let F(n) denote the number of well-formed, negated normal form formulas2 there are that use precisely n propositional variables exactly one time each. So F(1) = 2, F(2) = 16, and F(4) = 15360.(d)Using your answer for part (c), give  an expression for F(n). (4 marks)2Note: we do not assume ∧ and ∨ are associativeProblem 5 (18 marks)Consider the following graph:and consider the following process:Initially, start atv1.At each time step, choose one of the vertices adjacent to your current location uniformly at random, and move there.Boolean Algebra代写Let p1(n), p2(n), p3(n), p4(n) be the probability your location after n time steps is v1, v2, v3, or v4 respectively. So p1(0) = 1 and p2(0) = p3(0) = p4(0) = 0.(a)Expressp1(n + 1), p2(n + 1), p3(n + 1), and p4(n + 1) in terms of p1(n), p2(n), p3(n), and p4(n).(6 marks)Boolean Algebra代写(b)As n gets larger, each pi(n) converges to a single value (called the steady state) which can be deter- mined by setting pi(n + 1) = pi(n) in the above    Determine the steady state probabilities  forall vertices. (8 marks)(c)The distance between any two vertices is the length of the shortest path between them. What is your expected distance from v1in the steady state? (4 marks)Advice on how to do the assignmentAll submitted work must be done individually without consulting someone else’s solutions in accordance with the University’s “Academic Dishonesty and Plagiarism” policies.Boolean Algebra代写Assignmentsare to be submitted via WebCMS (or give) as a single pdfBe careful with giving multiple or alternative answers. If you give multiple answers, then we  will  give you marks only for your worst answer, as this indicates how well you understood the question.Boolean Algebra代写Some of the questions are very easy (with the help of the lecture notes or book). You can use the material presented in the lecture or book (without proving it). You do not need to write more than necessary (see comment above).When giving answers to questions, we always would like you to prove/explain/motivate your an- swers.Boolean Algebra代写If you use further resources (books, scientific papers, the internet, ) to formulate your answers, then add references to your sources.Boolean Algebra代写其他代写：考试助攻 计算机代写 java代写 algorithm代写 assembly代写 function代写paper代写 web代写 编程代写 report代写 algorithm代写 数学代写 finance代写 作业代写  python代写 code代写 Haskell代写合作平台：315代写 315代写 写手招聘 Essay代写

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