Suppose we have a recursive sequence J1,. For the purposes of this problem, it does not matter exactly how the fi are defined, only that they are recursively defined. For integer n ≥ 1, let P(n) be the predicate that i=1 fi = 2fn+2 - 3. Don't worry about whether this predicate "makes sense"; we haven't defined the fi so you won't be able to "make sense" of the P(n). It's not important for this problem. Consider a proof by induction that Vn ≥ 1: P(n). Suppose we've gotten to the inductive step, and suppose that the first steps of the inductive sten are

Suppose we have a recursive sequence J1,. For the purposes of this problem, it does not matter exactly how the fi are defined, only that they are recursively defined. For integer n ≥ 1, let P(n) be the predicate that i=1 fi = 2fn+2 - 3. Don't worry about whether this predicate "makes sense"; we haven't defined the fi so you won't be able to "make sense" of the P(n). It's not important for this problem. Consider a proof by induction that Vn ≥ 1: P(n). Suppose we've gotten to the inductive step, and suppose that the first steps of the inductive sten are

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Suppose we have a recursive sequence f1, f2, f3,.... For the purposes of this problem, it does not matter exactly how the fi are defined, only that they are recursively defined. For integer n ≥ 1, let P(n) be the predicate that fn = 2n². Don't worry about whether this predicate "makes sense"; we haven't defined the f; so you won't be able to "make sense" of the P(n). It's not important for this problem. Consider a proof by induction that Vn ≥ 1: P(n). Suppose we've gotten to the inductive step, and suppose that the first step of the inductive step is fk+1=2fk + k True or false: Based on the information given, we will need strong induction for this proof. True False
Suppose we have a recursive sequence f1, f2, f3,.... For the purposes of this problem, it does not matter exactly how the f; are defined, only that they are recursively defined. For integer n ≥ 1, let P(n) be the predicate that fn = 2n². Don't worry about whether this predicate "makes sense"; we haven't defined the f; so you won't be able to "make sense" of the P(n). It's not important for this problem. Consider a proof by induction that Vn 1: P(n). Suppose that the first step of the inductive step is fk+1 = (k − 1) · fk True or false: Based on the information given, we will need at least P(1), P(2) as base cases for this proof. True False
Question 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).
The three integers n, I and j, with I and j being between 1 and 2n, are the prerequisites to the issue. You have a board of squares that is 2n by 2n squares. Each tile has an adequate amount of pieces and the desired form. With the exception of the solitary square at position I j, you must cover every 2n 2n tile on the board by placing nonoverlapping tiles. Provide a recursive method for this issue where you lay one tile on your own and then enlist the aid of four buddies. a case is a phrase, right?
a. Correctness of dynamic programming algorithm: Usually, a dynamic programming algorithm can be seen as a recursion and proof by induction is one of the easiest way to show its correctness. The structure of a proof by strong induction for one variable, say n, contains three parts. First, we define the Proposition P(n) that we want to prove for the variable n. Next, we show that the proposition holds for Base case(s), such as n = 0, 1, . . . etc. Finally, in the Inductive step, we assume that P(n) holds for any value of n strictly smaller than n' , then we prove that P(n') also holds. Use the proof by strong induction properly to show that the algorithm of the Knapsack problem above is correct. b. Bounded Knapsack Problem: Let us consider a similar problem, in which each item i has ci > 0 copies (ci is an integer). Thus, xi is no longer a binary value, but a non-negative integer at most equal to ci , 0 ≤ xi ≤ ci . Modify the dynamic programming algorithm seen at class for this…
Give a recursive definition for the set of all strings of a’s and b’s that begins with an a and ends in a b. Say, S = { ab, aab, abb, aaab, aabb, abbb, abab..} Let S be the set of all strings of a’s and b’s that begins with a and ends in a b. The recursive definition is as follows – Base:... Recursion: If u ∈ S, then... Restriction: There are no elements of S other than those obtained from the base and recursion of S.
Let ∑={a,b} and T be the set of words in ∑* that have no consecutive a’s (i.e. there cannot be 2 or more a’s in a row). (a) Give a recursive definition for the set T. (b) Use your recursive definition to show that the string “abbaba” is in T. (c) Is your recursive definition uniquely determined? Explain why or why not.
Show the function f(i; k) is primitive recursive where f(i; k) = Pi.Pi+1............Pi+k. Recall, Pn is the nth prime number and P0 = 0.
Recursion can be direct or indirect. It is direct when a function calls itself and it is indirect recursion when a function calls another function that then calls the first function. To illustrate solving a problem using recursion, consider the Fibonacci series: - 1,1,2,3,5,8,13,21,34...The way to solve this problem is to examine the series carefully. The first two numbers are 1. Each subsequent number is the sum of the previous two numbers. Thus, the seventh number is the sum of the sixth and fifth numbers. More generally, the nth number is the sum of n - 2 and n - 1, as long as n > 2.Recursive functions need a stop condition. Something must happen to cause the program to stop recursing, or it will never end. In the Fibonacci series, n < 3 is a stop condition. The algorithm to use is this: 1. Ask the user for a position in the series.2. Call the fib () function with that position, passing in the value the user entered.3. The fib () function examines the argument (n). If n < 3…
1. Compute the recursive function f(n) = 3f(n–3)– l; ƒ(0) = 1; for n=12;. Show %3D %3D %3D clearly how the function is evaluated and the return of the function for every recursive step as seen in class. Show all your work to get credit
Let p be a proposition and P be a propositional function. Identify if the following statement is always true, never true, or only sometimes true/false: T→p⇒T^p Always True O Never True O Sometimes True Suppose there is a robot that builds a copy of itself in 2 hours. The copy then starts to build copies of itself as well. Let be the total number of fully functional robots after n hours. Suppose ro = 1. Recursive Case, 'n = 'n = [(n-2) +2 In = [(n-2) *2 In = [(n-1) +1 O None. Function description: g: Z→R g(x) = (x − 2)(x+2)x Identify if g is: 1. One-to-One II. Onto III. One-to-One Correspondence O IV. None
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