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22a-Counting

22a-Counting - Recursive Definitions and Introduction...

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Recursive Definitions and Counting Discrete Mathematics

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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 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 22-4 Well Formed Propositional Statements What is a well formed statement? (p q) ¬ r is well formed (p q) ¬ r is not Recursive definition of well formed formulas Basis step: A primitive statement is a well formed statement Inductive step: If Φ and Ψ are well formed statements, then ¬ Φ , ( Φ Ψ ), ( Φ Ψ ), ( Φ Ψ ), ( Φ Ψ ), ( Φ Ψ ) are well formed statements Such a definition can be used by various algorithms, for example, parsing
Discrete Mathematics - Recursive Definitions and Counting 22-5 Fractals Fractals are curves defined recursively Basis step: Fractal of level 0 is just a segment Inductive step: Divide every segment of the fractal of level k into 3 equal parts and remove the middle one. Insert in this place two sides of a equilateral triangle

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Discrete Mathematics - Recursive Definitions and Counting 22-6 Rooted Trees A binary tree is a graph formed by the following recursive definition Basis case:
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