a) Give a recursive deﬁnition for the set of all strings of a’s and b’s where n a’s followed by n b’s (where n= 0, 1, 2, 3, ...). (Assume, S is set of all strings of a’s and b’s. Then S = {λ, ab, aabb, aaabbb, ... ) b) Give a recursive deﬁnition for the set of all strings of 'a' 's and 'b' 's that ends with an 'ab'. (Assume, S is set of all strings of a’s and b’s. Then S = {ab, aab, bab, abab, baab, aaab, bbab, ... )

a) Give a recursive deﬁnition for the set of all strings of a’s and b’s where n a’s followed by n b’s (where n= 0, 1, 2, 3, ...). (Assume, S is set of all strings of a’s and b’s. Then S = {λ, ab, aabb, aaabbb, ... ) b) Give a recursive deﬁnition for the set of all strings of 'a' 's and 'b' 's that ends with an 'ab'. (Assume, S is set of all strings of a’s and b’s. Then S = {ab, aab, bab, abab, baab, aaab, bbab, ... )

C++ Programming: From Problem Analysis to Program Design

8th Edition

ISBN:9781337102087

Author:D. S. Malik

Publisher:D. S. Malik

Chapter15: Recursion

Section: Chapter Questions

Problem 6PE

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Q4. Recursion Find how many possible combinations that a number can be decomposed into the multiple of integers (smaller than the number itself) by a recursion function.Suppose we have a positive number Y as input, and it need to be decomposed into the multiple of several integers, Yi, where each Yi, is smaller than Y(e.g., Y=Y1∗Y2∗ Y3∗...∗Yn). In addition, these decomposed integers can only be arranged in an ascending order (Y1
A palindrome is a sentence that contains the same sequence of letters reading it either forwards or backwards. A classic example is '1\.ble was I, ereI saw Elba." Write a recursive function that detects whether a string is apalindrome. The basic idea is to check that the first and last letters of thestring are the same letter; if they are, then the entire string is a palindromeif everything between those letters is a palindrome.There are a couple of special cases to check for. If either the first orlast character of the string is not a letter, you can check to see if the restof the string is a palindrome with that character removed. Also, when youcompare letters, make sure that you do it in a case-insensitive way.Use your function in a program that prompts a user for a phrase andthen tells whether or not it is a palindrome. Here's another classic fortesting: '1\. man, a plan, a canal, Panama!"
Question#1: Let P and Q be integers and suppose F (P, Q) is recursively defined by: F(int P, int Q) if (P == 0) return Q else return F(P- 1, P + Q) Find and Trace F(8,1)
For function decToBinary, write the missing parts of the recursion case. This function should return a string that stores the binary equivalent for int variable num. Example: The binary equivalent of 13 may be found by repeatedly dividing 13 by 2. So, 13 in base 2 is represented by the string "1101". Examples: decToBinary(13) -> "1101" public String decToBinary (int num) { if (num < 2) return Integer.toString(num); else return <<Missing recursive call>> + <<Missing calculation>>;}
1. Write a recursive function named RecursiveSum that, given a positive number n (n> 0), returns the sum of the numbers from 1 to n. Also, show the process of finding the result using a recursion tree.
please code in python Write a recursive function reverse(sentence) for reversing a sentence. For example, reverse('Who let the dogs out?') will return '?tuo sgod eht tel ohW'. The idea is to remove the first or last letter, reverse the shortened sentence, and then combine the two parts.
Write recursive methods for the following operations: • removeDup(String s): takes string s and returns another string r constructed from s after removing all duplicates. For example, when s = "pineapples", r will be "pineals". • intersection(String s1, String s2): takes two duplicate-free strings and returns the string consisting of all letters that appear in both s1 and s2. • union(String s1, String s2): takes two duplicate-free strings and returns the string consisting of all letters that appear in either s1 or s2. • difference(String s1, String s2): takes two duplicate-free strings and returns the string consisting of all letters that appear only in s1. For example, when s1 = "aples" and s2 = "pears", intersection(s1, s2) will return "apes", union(s1, s2) will "aplesr", and difference(s1, s2) "l". Calling method removeDup() is not allowed in intersection(s1, s2) , union(s1, s2) or difference(s1, s2) method. Your program should check strings are duplicate free before they are…
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…
Algortimhs Recursive Thinking) Carefully read the comment of the following function, then complete the function so that it correctly solves the problem. Verify your answer by writing down the three questions rule and answer the questions for your completed algorithm. /* Return true if there exist duplicate values in sublist a[first...last], i.e., there are a[i]==a[j] for some i,j where i is not equal to j @param a: the vector/list @param first, last: two integer specifying the starting and ending indices of list@return return true if there are duplicate values in a[first...last], otherwise return false */ bool ContainDuplicate (vector<int> a, int first, int last) { //1. what should be the base case? What should be returned? // general case if (ContainDuplicate (a, first+1, last)==true) _______________; //Finish the rest based upon comment:// if a[first] does not equal to ANY value in a[first+1...last], // then return false;// else return true } (Recursive Thinking) For the…
Pascal's triangle is a useful recursive definition that tells us the coefficients in the expansion of the polynomial (x + a)^n. Each element in the triangle has a coordinate, given by the row it is on and its position in the row (which you could call a column). Every number in Pascals triangle is defined as the sum of the item above it and the item above it and to the left. If there is a position that does not have an entry, we treat it as if we had a 0 there. *picture of the pascals triangle* Given the following recursive function signature, write the recursive function that takes a row and a column and finds the value at that position in the triangle. Assume that the triangle starts at row 0 and column 0. Examples: pascal(2, 1) -> 2, pascal(1, 2) -> 0 public int pascal(int row, int column) { }
Need help with this c++ recursive function: Write the function starCrossed() which computes (recursively, with no loops) a new string where identical characters that are adjacent in the original string are separated from each other by a "*". Here are some examples: starCrossed("hello") returns "hel*lo" starCrossed("xxyy") returns "x*xy*y" starCrossed("aaaa") returns "a*a*a*a"
The 4th problem mimics the situation where eagles flying in the sky can be spotted and counted.FindEagles: a recursive function that examines and counts the number of objects (eagles) in aphotograph. The data is in a two-dimensional grid of cells, each of which may be empty (value 0) orfilled (value 1 to 9). Maximum grid size is 50 x 50. The filled cells that are connected form an object(eagle). Two cells are connected if they are vertically, horizontally, or diagonally adjacent. Thefollowing figure shows 3 x 4 grids with 3 eagles. 0 0 1 21 0 0 01 0 3 1 FindEagle function takes as parameters the 2-D array and the x-y coordinates of a cell that is a part ofan eagle (non-zero value) and erases (change to 0) the image of an eagle. The function FindEagleshould return an integer value that counts how many cells has been counted as part of an eagle and havebeen erased. The following sample data has two pictures, the first one is 3 x 4, and the second one is 5 x 5 grids. Notethat your program…