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CUSTOMER_CODE SMUDE DIVISION_CODE SMUDE EVENT_CODE Jan2017 ASSESSMENT_CODE MC0082_Jan2017 QUESTION_TYPE DESCRIPTIVE_QUESTION QUESTION_ID 5146 a.Explain the five ways to describe a set. QUESTION_TEXT b.What is the value of c.Define Deterministic Finite Automata. a.(5 marks) i.Describe a set by describing the properties of the members of the set. ii.Describe a set by listing its elements. iii.Describe a set A by its characteristic function. iv.Describe a set by recursive formula. This is to give one or more elements of the set and a rule by which the rest of the elements of the set may be generated. v.Describe a set by an operation (say union, intersection, complement etc ) on some other set. SCHEME OF EVALUATION b.=(1/2)+(2/3)+(3/4)=23/12 (2 marks) c.A DFA is 5-tuple or quintuple M=(Q, Σ, δ, q 0, F) where Q is non-empty, finite set of states Σ is non-empty, finite set of input alphabet Δ is transition function, which is mapping from Q x Σ to Q. for this transition function the parameters to be passed are state and input symbol. Based on the current state and input symbol, the machine may enter into another state. q 0∈Q is the start state. F⊆Q is set of accepting or final state. (3 marks) QUESTION_TYPE DESCRIPTIVE_QUESTION QUESTION_ID 5147 QUESTION_TEXT Explain the concept of mathematical induction. Let A be the set of all natural numbers such that i.0∈A, and ii.for each natural number n, if {0, 1, … , n}⊆A, then n+1 ∈A Then A = N SCHEME OF EVALUATION In other words: The principle of mathematical induction states that any set of natural numbers containing zero and with the property that it contains n+1 whenever it contains all the numbers up to and including n, must in fact be the set of all natural numbers. In practice, induction is used to prove the assertions of the following form: “For all natural numbers n, property P is true.” The above principle is applied to the set A = {n:P is true of n} in the following way. a.In the basis step we show that 0∈A, i.e. P is true of 0. b.The induction hypothesis is the assumption that for some fixed but arbitrary n≥0, P holds for each natural number 0, 1, 2, … ,n. c.In the induction step we show, using the induction hypothesis, that P is true of n+1. By the induction principle, A is then equal to N, that is P holds for every natural number. QUESTION_TYPE DESCRIPTIVE_QUESTION QUESTION_ID 72481 QUESTION_TEXT Obtain a PDA to accept the language L={anb2n|n≥1}. SCHEME OF EVALUATION The PDA to accept the given language is M = (Q, ∑, I’, δ, Z0, F) where Q = {q0, q1, q2}, q0 is the start state ∑ = {a, b} Г = {a, Z0}, is the initial number of the stack, F= {q2} is the final state. 5 Marks The transitions δ are given by 5 Marks δ(q0, a, Z0) = (q0, aaZ0) δ(q0, a, a) = (q0, aaa) δ(q0, b, a) = (q1, Λ) δ(q1, b, a) = (q1, Λ) δ(q1, Λ, Z0) = (q2, Λ) QUESTION_TYPE DESCRIPTIVE_QUESTION QUESTION_ID 109984 QUESTION_TEXT Explain Big Omega and Big Theta function with example. Big Omega Description (3 marks) Example (2 marks) SCHEME OF EVALUATION Big Theta Description (3 marks) Example (2 marks) QUESTION_TYPE DESCRIPTIVE_QUESTION QUESTION_ID 109985 QUESTION_TEXT Use the definition of order to show that, 5x 3 – 3x+4 is O(x 3) The functions f and g referred to the definition of O-notation are defined as follows. For all real numbers For all real numbers x > 1, and (by the triangle inequality) SCHEME OF EVALUATION Therefore, or for all x > 1. for all x > k where C = 12 and k = 1. Hence, QUESTION_TYPE DESCRIPTIVE_QUESTION QUESTION_ID 109989 QUESTION_TEXT Briefly explain 2 recursive functions with example. Recursive theorem SCHEME OF EVALUATION Fibonacci sequence GCD (Any 2 of the above 5 marks each)