Given the following recursively defined set S: Basis: 0 € S and 7 € S Recursive rule: if x ES and y = S, then: • x+yЄS x-yes Prove that every element in S is divisible by 7. . . (3 pt.) Complete the basis step of the proof. (3 pt.) What is the inductive hypothesis? • (3 pt.) What do you need to show in the inductive step of the proof? . (6 pt.) Complete the inductive step of the proof.
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- Let the set S for concatenating binary string be defined recursively by Basis: A, 1 E S Recursion: If w ES then OwES and w0 ES. The following diagram is not a proper representation of 4 recursive calls. Initial Condition Apply recursive step W 0w Ww0 W0w WW0 OX=0 20-0 01 10 equal Apply recursive step 00 00 001 010 010 100 Apply recursive step 000 000 0001 (0010 0010 0100 0100) 1000 O True O FalseQuestion 4: The function T(n) is recursively defined as follows: 1 if n = 1, 고 +. T(n -1) if n> 2. Prove that T(n) = 0(n log n).8- Determine if each of the following recursive definition is a valid recursive definition of a function f from a set of non-negative integers. If f is well defined, find a formula for f(n) where n is non- negative and prove that your formula is valid. a. f(0) = 2,f(1) = 3, f(n) = f(n-1)-1 for n ≥ 2 b. f(0) = 1,f(1) = 2, f(n) = 2f (n-2) for n = 2
- S is a set of strings over the alphabet {a, b}* recursively defined as: Base case: λ ∈ S, a ∈ S, b ∈ S Recursive rules: If x ∈ S, then Rule 1: axb ∈ S Rule 2: bxa ∈ S List all the strings in S of length 3.a) Suppose the following recursive set S: • Basis elements: {0, 2, 4} • Recursive step 1: r, y E S → * y € S • Recursive step 2: a, y, z E S –→ x + y + z € S Prove, by structural induction, that all elements in S are even.Construct a DFA A so that L(A) = L(N) where N is the following NFA:
- Question: Let t(x) be the number of primes that areGiven three sequences of length m, n, and p each, you are to design and analyze an algorithm to find the longest common subsequence (LCSS) for the three sequence. Is it possible to use dynamic programming to find the LCSS between the three sequences? If yes, provide recursive solution to the above problem. (Hint: We already designed dynamic programming algorithm for two sequences of length m and n in our class.)T/F 8. Without recursion, the Koch snowflake = 1 could be drawn.The following recursive definition of a set S over {a, b, c, d}. Basis: "", a, b, c E S. Recursive Step: If ax E S, then bax ES and cax E S. If bx € S, then cbx € S. If cx = S, then bax ES and bcx = S. If dx E S, then ddx E S. Closure: SE S only if it is a, b or c or it can be obtained from a, b or c using finitely many operations of the Recursive Step. a) What recursive step can be removed from the above definition and why? b) Derive from the definition of S that bcbc S. c) Explain why baa & S.7. A game is played by moving a marker ahead either 2 or 3 steps on a linear path. Let cn be the number of different ways a path of length n can be covered. Given, Cn =Cn-2+ Cn-3, C1=0, c2-D1, c3=1 Write a recursive algorithm to compute Cn-Please do not give solution in image format thanku Implement the following recursively using sudo code: Suppose an elevator which is on a floor on n. For this elevator to go from the nth floor to the base(ground) floor, it should go to every floor under the nth floor. Let's consider an elevator on the 4th floor. This elevator 1st comes on the third(3rd) floor. 4th-> 3rd then 3rd-> 2nd then 2->1 and next 1->0(ground floor) The recursive equation defined is F(n)=1+F(n-1)SEE MORE QUESTIONS