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Note that if we don't have the Extremal Clause,  0.5, 1.5, 2.5, ... can be included in N, which is not what we want as the set of natural numbers.

Example 2. Definition of the Set of Nonnegative Even Numbers NE

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

Basis Clause: 0 ∈ NE

Inductive Clause: For any element x in NE, x + 2 is in NE.

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

Example 3. Definition of the Set of Even Integers EI

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

Basis Clause: 0 ∈ EI

Inductive Clause: For any element x in EI, x + 2, and x - 2 are in EI.

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

Example 4. Definition of the Set of Strings S over the alphabet {a,b} excepting empty string. This is the set of strings consisting of a's and b's such as abbab, bbabaa, etc.

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

Basis Clause: a ∈ S, and b ∈ S.

Inductive Clause: For any element x in S, ax ∈ S, and bx ∈ S.

Here ax means the concatenation of a with x.

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

Tips for recursively defining a set:

For the "Basis Clause", try simplest elements in the set such as smallest numbers (0, or 1), simplest expressions, or shortest strings. Then see how other elements can be obtained from them, and generalize that generation process for the "Inductive Clause".

The set of propositions (propositional forms) can also be defined recursively.

Generalized set operations

As we saw earlier, union, intersection and Cartesian product of sets are associative. For example (A ∪ B) ∪ C = A ∪ (B ∪ C)

To denote either of these we often use A ∪ B ∪ C.

This can be generalized for the union of any finite number of sets as A1 ∪ A2 ∪.... ∪ An.

which we write as

       i = 1 n A i size 12{ union rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } } {}

This generalized union of sets can be rigorously defined as follows:

Definition ( i = 1 n A i size 12{ union rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } } {} ):

Basis Clause: For n = 1, i = 1 n A i = A 1 size 12{ union rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } =A rSub { size 8{1} } } {} .

Inductive Clause:   i = 1 n + 1 A i size 12{ union rSub { size 8{i=1} } rSup { size 8{n+1} } A rSub { size 8{i} } } {} = i = 1 n A i size 12{ union rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } } {} ∪ An+1

Similarly the generalized intersection i = 1 n A i size 12{ intersection rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } } {} and generalized Cartesian product i = 1 n A i size 12{ times rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } } {} can be defined.

Based on these definitions, De Morgan's law on set union and intersection can also be generalized as follows:

Theorem (Generalized De Morgan)

i = 1 n A i ¯ = i = 1 n A i ¯ size 12{ {overline { union rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } }} = intersection rSub { size 8{i=1} } rSup { size 8{n} } {overline {A rSub { size 8{i} } }} } {} ,     and

i = 1 n A i ¯ = i = 1 n A i ¯ size 12{ {overline { intersection rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } }} = union rSub { size 8{i=1} } rSup { size 8{n} } {overline {A rSub { size 8{i} } }} } {}

Proof: These can be proven by induction on n and are left as an exercise.

Recursive definition of function

Some functions can also be defined recursively.

Condition: The domain of the function you wish to define recursively must be a set defined recursively.

How to define function recursively: First the values of the function for the basis elements of the domain are specified. Then the value of the function at an element, say x, of the domain is defined using its value at the parent(s) of the element x.

A few examples are given below.

They are all on functions from integer to integer except the last one.

Example 5: The function f(n) = n! for natural numbers n can be defined recursively as follows:

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Source:  OpenStax, Discrete structures. OpenStax CNX. Jan 23, 2008 Download for free at http://cnx.org/content/col10513/1.1
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