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22h - Discrete Mathematics Recursive Definitions and...

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Introduction Recursive Definitions and Counting Discrete Mathematics Andrei Bulatov Discrete Mathematics - Recursive Definitions and Counting 22-2 Previous Lecture Strong induction Induction and well ordering Recursively defined functions Discrete Mathematics – Recursive Definitions and Counting 22-3 Recursively Defined Functions Induction mechanism can be used to define things. To define a function f: N R we complete two steps: Basis step: define f(1) Inductive step: For all k define f(k + 1) as a function of f(k), or, more general, as a function of f(1), f(2), … , f(k). Give a recursive definition of f(n) = Basis step: f(0) = 1 Inductive step: f(k + 1) = 2 f(k). n 2 Discrete Mathematics – Recursive Definitions and Counting 22-4 Factorial Another useful recursively defined function is factorial f(n) = n! Basis step: 0! = 1 Inductive step: (k + 1)! = k! (k + 1) n n! 0 1 2 3 4 1 1 2 6 24 n n! 5 6 7 8 9 120 720 5040 40320 362880 Discrete Mathematics – Recursive Definitions and Counting 22-5 Fibonacci Numbers Usually, Fibonacci numbers are thought of as a sequence of natural numbers, but as we know such a sequence can also be viewed as a function from N . F(n) Basis step: F(1) = F(2) = 1 Inductive step: F(k + 1) = F(k) + F(k – 1) n 1 2 3 4 5 6 7 8 9 10 11 12 13 F(n) 1 1 2 3 5 8 13 21 34 55 89 144 233 Binet’s formula 5 ) 1 ( ) ( n n n F ϕ ϕ - - = where ϕ is the golden ratio 49 6180339887 . 1 2 5 1 = ϕ Discrete Mathematics - Recursive Definitions and Counting 22-6 Recursively Defined Sets and Structures Induction can be used to define structures We need to complete the same two steps: Basis step: Define the simplest structure possible Inductive step: A rule, how to build a bigger structure from smaller ones.
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Discrete Mathematics - Recursive Definitions and Counting 22-7 Well Formed Propositional Statements What is a well formed statement?
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