# 0.5 Discrete structures recursion

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## Recursive definition

Sets which have too many elements to list them up, and for which there are no convenient or obvious predicates to specify their elements can often be defined using a recursive definition (also called inductive definition). It essentially gives a procedure to generate the members of the set one by one starting with some subset of its elements. In this type of definition, first a collection of elements to be included initially in the set is specified. These elements can be viewed as the seeds of the set being defined. Next, the rules to be used to generate elements of the set from elements already known to be in the set (initially the seeds) are given. These rules provide a method to construct the set element by element starting with the seeds. These rules can also be used to test elements for the membership in the set.

A recursive definition of a set always consists of three distinct clauses:

1. The basis clause (or simply basis) of the definition establishes that certain objects are in the set. This part of the definition specifies the "seeds" of the set from which the elements of the set are generated using the methods given in the inductive clause. The set of elements specified here is called basis of the set being defined.

2. The inductive clause (or simply induction) of the definition establishes the ways in which elements of the set can be combined to produce new elements of the set. The inductive clause always asserts that if objects are elements of the set, then they can be combined in certain specified ways to create other objects. Let us call the objects used to create a new object the parents of the new object, and the new object is their child.

3. The extremal clause asserts that unless an object can be shown to be a member of the set by applying the basis and inductive clauses a finite number of times, the object is not a member of the set.

The set you are trying to define recursively is the set that satisfies those three clauses.

There are a number of other ways of expressing the extremal clause that are equivalent to the extremal clause given above.

Examples of Recursive Definition of Set

Example 1. Definition of the Set of Natural Numbers N

The set N is the set that satisfies the following three clauses:

Basis Clause: 0 ∈ N

Inductive Clause: For any element x in N, x + 1 is in N.

Extremal Clause: Nothing is in N unless it is obtained from the Basis and Inductive Clauses.

The basis for this set N is { 0 } . The x + 1 in the Inductive Clause is the parent of x, and x is the child of x + 1. Following this definition, the set of natural numbers N can be obtained as follows:

First by the Basis Clause,   0 is put into N. Then by the Inductive Clause, since 0 is in N,  0 + 1 (= 1) is in N. 0 is the parent of 1, and 1 is the child of 0. Then by the Inductive Clause again,   1 + 1 (= 2) is in N. 1 is the parent of 2, and 2 is the child of 1. Proceeding in this manner all the "natural numbers" are put into N.

Introduction about quantum dots in nanotechnology
what does nano mean?
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Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
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it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
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Do somebody tell me a best nano engineering book for beginners?
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what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
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is Bucky paper clear?
CYNTHIA
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so some one know about replacing silicon atom with phosphorous in semiconductors device?
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Harper
Do you know which machine is used to that process?
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how to fabricate graphene ink ?
for screen printed electrodes ?
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What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
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China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
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many many of nanotubes
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what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
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what is system testing?
how did you get the value of 2000N.What calculations are needed to arrive at it
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