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are wellformed 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 wellformed 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 (n1)! 14 Example 2: Fibonacci Numbers
The Fibonacci numbers fn are defined as
follows:
Base step: f0=0 and f1=1
Inductive step: fn = fn1 + fn2 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.
 Fall '11
 math
 Recursion

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