Examples 42 x x42 xy 30 xyz 8 example

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Unformatted text preview: nd (F/G) are well-formed formulas. Examples: 42, x, (x+42), (x/y), (3/0), (x*(y+z)) 8 Example 4: Lists We can define the set L of finite lists of integers as follows. Basis: The empty list () is contained in L Induction: If i is an integer, and l is a list in L, then (cons i l) is in L. [Note: This is the Lisp style of lists, where (cons i l) appends the data item i at the front of the list l] Example: (cons 1 (cons 2 (cons 3 () ))) is the list (1 2 3) in Lisp. 9 Example 5: Binary Trees We can define the set B of binary trees over an alphabet A as follows: Basis: ⟨⟩∈B. Induction: If L, R∈B and x∈A, then ⟨L,x,R⟩∈B. Example: ⟨⟨⟩,1,⟨⟩⟩ // tree with one node (1) Example: ⟨ ⟨⟨⟩,1,⟨⟩⟩, r, ⟨⟨⟩,2,⟨⟩⟩ ⟩ // tree with root r and two children (1 and 2). 10 Applications of Inductively Defined Sets In Computer Science, we typically use inductively defined sets (a.k.a. recursively defined sets) when defining: - programming languages (via grammars) - logic (via well-formed logical formulas) - data structures (binary trees, rooted trees, lists). - fractals We also use them in connection with functional programming languages. Extremely popular in Computer Science! 11 Recursively Defined Functions 12 Recursively Defined Functions Suppose we have a function with the set of nonnegative integers as its domain. We can specify the function as follows: Basis step: Specify the value of the function at 0 Inductive step: Give a rule for finding its value at an integer from its values at smaller integers. This is called a recursive or inductive definition. 13 Example 1: Factorial Function We can define the factorial function n! as follows: Base step: 0! = 1 Inductive step: n! = n (n-1)! 14 Example 2: Fibonacci Numbers The Fibonacci numbers fn are defined as follows: Base step: f0=0 and f1=1 Inductive step: fn = fn-1 + fn-2 for n >=2 We can use the recursive definition of the Fibonacci numbers to prove many properties of these numbers. The recursive structure actually helps to formulate the proofs. 15 Recursively Defi...
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This note was uploaded on 03/24/2014 for the course CSCE 222 taught by Professor Math during the Fall '11 term at Texas A&M.

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