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Recursively Defined Functions - Discrete Mathematics - Lecture Slides

Slides, Discrete Mathematics

Post: April 27th, 2013
Description
During the study of discrete mathematics, I found this course very informative and applicable.The main points in these lecture slides are:Recursively Defined Functions, Recursive Step, Fibonacci Sequence, Positive Integer, Recursively Defined Sets, Recursively Defined Languages, Recursively Defined Structures, Binary Trees, Fibonacci Proof, Inductive Proof Process
During the study of discrete mathematics, I found this course very informative and applicable.The main points in these lecture slides are:Recursively Defined Functions, Recursive Step, Fibonacci Sequence, Positive Integer, Recursively Defined Sets, Recursively Defined Languages, Recursively Defined Structures, Binary Trees, Fibonacci Proof, Inductive Proof Process
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Discrete Mathematics CS 2610 October 21, 2008 Docsity.com Recursively Defined Functions We are familiar with the use of formulae to define functions. But it is also possible to define some functions using recursion. Example: f(n) = n2 for n ∈ Z≥0 or… f(0) = 0 f(n+1) = f(n) + (2n + 1), for n ∈ Z≥0 2 Docsity.com Recursively Defined Functions Defining a function recursively requires a basis step (or steps) and a recursive step, just like induction. Example: f(n) = n!, for n ∈ Z≥0  Basis: f(0) = 1 Yup, that’s the def. of 0!  Rec. Step: f(n+1) = f(n) ⋅ (n+1), n ∈ Z≥0 Example: Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, …   Basis: f(0) = 0, f(1) = 1 Rec. Step: f(n+2) = f(n+1) + f(n), for n ∈ Z≥0 3 Docsity.com Recursively Defined Sets We can define sets in the same manner, by establishing a basis and recursive step. Example: S = { x | x is an odd, positive integer }   Basis: Rec. Step: 1∈S if x ∈ S, then x + 2 ∈ S 4 Docsity.com ..

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