Give a recursive definition for the set of all strings of 0’s and 1’s that have either twice the number of 0's as 1's or twice the number of 1's as 0's.

Write a C++ program using classes and recursion functions to covert a number in a given base (the user enters the base - from 2 to 9) to a decimal number. The algorithm states that each successive digit in the number is multiplied by the base raised to the power corresponding to its position in the number. The low-order digit is in position 0. We sum together all of these products to get the decimal value. For example, in binary- base 2, if we have the binary number 111001, we convert it to decimal as follows: 1x25 + 1x2* + 1x23 + 0x22+ 0x22 + 1x2o = 57 For a base 5 number, 14230: 1x5 + 4x53 + 2x52 + 3x5+ Ox5° = 1190 Validate your input. You should check for invalid digits and provide an informative error message to the user. This program requires you to use a recursive function for the conversion to decimal. Failure to do that will result in no points awarded.

Consider a recursive function, called f, that computes powers of 3 using only the + operator. Assume n> = 0. int f(int n) { if (n == 0) return 1; return f(n-1) + f(n-1) + f(n-1); } Give an optimized version of f, called g, where we save the result of the recursive call to a temporary variable t, then return t+t+t. i got int g(int n) { if (n == 0) return 1; int t = g(n - 1); return t+t+t; } so now Write a recurrence relation for T(n), the number addition operations performed by g(n) in terms of n.

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.

Solve the recursion by substitution: T(n) = 3T(n/2)+ 5n where T(1)=1

In kotlin, Write a recursive function with an expression body that takes a list of Pairs in which each value may be of any type, and returns a list of the first elements from each pair.

In Kotlin, Write a recursive function called myZip, with an expression body. MyZip takes two Lists and returns a List of Pairs in which each value may be any type (I suggest your start by thinking out how to represent this data type). The pairs consist of corresponding elements in the two lists (the first element of the first list and the first element of the second list, etc). The base case should be that either (or both) of the original lists has length 1, so that, if the lists have different length, the zipping stops when the first list runs out of values.

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.

Write a recursive Racket function "update-if" that takes two functions, f and g, and a list xs as parameters and evaluates to a list. f will be a function that takes one parameter and evaluates true or false. g will be a function that takes one parameter and evaluates to some output. The result of update-if should be a list of items such that if x is in xs and (f x) is true, then (g x) is in the list. The output list's elements should keep the input list's items in the same relative order. For example (update-if even? add1 '(1 2 3 4)) should evaluate to '(3 5) because 2 and 4 make even? true and add1 turns 2 and 4 into 3 and 5.

solve by hand do not use code