Recursively Defined Functions - Discrete Mathematics - Lecture Slides, Slides for Discrete Mathematics. Islamic University of Science & Technology

Discrete Mathematics

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
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Discrete Mathematics
CS 2610
October 21, 2008
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.
f(n) = n2 for n Z≥0
f(0) = 0
f(n+1) = f(n) + (2n + 1), for n Z≥0
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
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
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: 1 S
Rec. Step: if x S, then x + 2 S
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