Sequences, Induction, and Recursion

Sequences, Induction, and Recursion - Math 100 Recursion...

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Unformatted text preview: Math 100 Recursion and Induction Proofs Martin H. Weissman 1. Talking about sequences The set of natural numbers N is defined informally by the following: N = { , 1 , 2 , 3 ,... } . Note that some people have the natural numbers start at 1, instead of 0. In this class, we always define N to include 0. There are some basic properties of natural numbers which you may assume these are the Peano axioms: (1) Zero is a natural number. (Symbolically, 0 N ). (2) If n is a natural number, then n + 1 is a natural number. (3) There is no natural number n , such that n + 1 = 0. (4) If m and n are natural numbers, and m + 1 = n + 1, then m = n . A sequence is simply a list, which is indexed by natural numbers: a ,a 1 ,a 2 ,a 3 ,.... We can talk about sequences of real numbers, sequences of integers, sequences of sets, etc..., even sequences of sequences, on occasion. Here are three examples of sequences, given informally : 1 , 1 , 2 , 3 , 5 , 8 , 13 ,.... , 10 , 1110 , 3110 , 132110 , 1113122110 ,.... 1 , 1 / 2 , 1 / 3 , 1 / 4 , 1 / 5 ,.... 2 , 2 , q 2 , r q 2 ,.... , {} , { , {}} , { , { , {}}} ,.... True,False,True,False,True,False,.... A sequence is most often given by one-term recursion: A zeroth term. A rule, which states how each term can be computed from the previous term. Sometimes, a sequence is given by two-term recursion: The first two terms are given. A rule is given, which states how each term can be computed from the previous two terms. Most generally, a sequence can be given by full recursion: The first k terms are given, for some positive integer k . A rule is given, which states how each term can be computed from none, some, or all of the previous terms. Defining such sequences relies on the following axiom of induction : Given data for recursion (one-term, two-term, or full), there is a unique se- quence which satisfies that rule....
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Sequences, Induction, and Recursion - Math 100 Recursion...

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