Induction Proof

If n >= 1, the number of strings using the digits 0,1, and 2 with no two consecutive places holding the same digit, is 3x2^n-1. For example, there are 12 such strings of length three: 010, 012, 020, 021, 101, 102, 120, 121, 201, 202, 210, and 212. Prove this claim by induction on the length of the strings. Is the formula true for n=0?

Induction Problem

Attached in word document Numbers of the form tn = n (n+1) / 2 are called triangular numbers, because marbles arranged in an equilateral triangle, n on a side, will total... (see attachment)

Error Correcting Code

If no two strings in a code differ in fewer than three positions, the we can actually correct a single error, by finding the unique string in the code that differs from the received string in only one position. It turns out that there is a code of 7-bit strings that corrects single errors and contains 16 strings. Find such a code. Hint: Reasoning it out is probably best, but if you get stuck, write a program that searches for the string. From the example in the book - the strings would be like ASCII sequence of bits - whereby ASCII has a seven bit code and then a parity bit placed on the front for error detecting.

Recursive definition

We can define sorted lists of integers as follows: BASIS - A list consisting of a single integer is sorted. INDUCTION - If L is a sorted list in which the last element is a and if b >= a, then L followed by b is a sorted list. Prove that this recursive definition of "sorted list" is equivalent to our original, nonrecursive definition, which is that the list consist of integers a1 <= a2 <= .... <= an Remember, you need to prove to parts: (a) If a list is sorted by the recursive definition, then it is sorted by the nonrecursive definition (b) if a list is sorted by the nonrecursive definition, then it is sorted by the recursive definition. Part(a) can use induction on... click for more

Induction by recursion

Define recursively the set of even-parity strings, by induction on the length of the string. Hint: It helps to define two concepts simultaneously, both the even-parity and odd-parity strings.