Recursive Definitions: Sets and Functions - Prof. Dali Wang, Study notes of Discrete Structures and Graph Theory

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Recursive

Section 3.4 Recursive Definitions

Recursive Definitions Recursive form „^ Recursive form

defines a set, an equation,

or a process by defining a starting set orvalue and giving a rule for continuing to buildthe set, equation, or process based onpreviously defined items.

Recursive Definitions Recursive Definition „^ Recursively Defined Functions^ Function,

F , defined on nonnegative integers. To give a recursive definition of

F :

„^ (Basis)

Specify

F (0). „^ (Recursive Step)

Give a rule for defining F ( n +1)

from^ F

evaluated at smaller values.

Recursive Definitions Example „^ Example (set):A recursive definition of N (set of naturalnumbers)1.^ Basis: 0 is in N (0 is the BBB).2.^ Induction: if n is in N then so is n + 1 (how to build newobjects from old: “add one to an old object toget a new one”).

Recursive Definitions Example „^ Example (function):A recursive definition of the

Fibonacci sequence

(Classical): A young pair of rabbits (one for each sex) isplaced on a island. After they are 2 months old, each pairproduces another pair each month. The number of pairsof rabbits after n month is f(n).1.^ Basis: f(0) = f(1) = 1(two initial conditions)2.^ Induction: f(n + 1) = f (n) + f(n - 1)(the recurrence equation).

Recursive Definitions Example „^ Example (set):A recursive definition of the set of strings over a finitealphabet.An^ alphabet

Σ^ is a finite set. A^ string

over^ Σ^

is a finite sequence of symbols from

The set of all strings over

Σ^ is denoted by

The^ empty string

(the string containing no symbols) is the string containing no symbols. Its length is 0 and it isdenoted by

λ. (Note

λ∈ Σ ***** .) wx^ ∈ Σ *****

whenever w

∈ Σ *****^ and x

A^ language

over^ Σ^ is a subset of

Recursive Definitions Example „^ Example: Given the recurrence relation a

n=2an-1^ -a

n-2^ , for n=2, 3,

4, … Determine if sequence {a

n^ } is a solution if,

a=3nn

Recursive Definitions ExampleCompound Interest:On your 21

st^ birthday you get a letter informing you that on

the day you were born, an eccentric rich aunt deposited$100,000 in a bank account earning 4% interestcompounded annually and she now intends to turn theaccount to you (provided that you can figure out thequestions bellow).1). Find out the worth of the account k year after you wereborn using a recursive definition.2). Find a closed-form representation of the account worth(for^ k th year) and prove that it is valid.