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We motivate the use of relations, as a way to encapsulate the information previously spreadacross a swath of propositional variables.

Relations: building a better (representation of) waterworld

So far, we have represented WaterWorld boards using propositions like A-has-2 and B-unsafe . You've probably already felt that this is unwieldy, having hundredspropositional variables running around,with only our naming convention implying any relation between them.Worse, this zoo of propositions doesn't reflect how we actually think about WaterWorld.For instance, the only way the rules recognize that locations A and B are near each other is because of several axioms which simultaneously involve A-has-2 and B-unsafe , etc. , in just the right way to result in our idea of the conceptneighbor. In fact, there is no way of talking about the location A directly; we only had propositions which dealt with its properties, such aswhether or not it neighbored exactly two pirates.

If writing a program about WaterWorld, our program should reflect our conception of the problem.However, as it stands, our conception corresponds to having many many Boolean variables named A-has-2 , B-unsafe , etc. Even worse, the rules would be encodings of the hundreds of axioms. A long enumeration of the axioms is probably not how you think of the rules.In other words, when explaining the game to your friend, you probably sayif a location contains a 2, then two of its neighbors are pirates, rather than droning on for half an hour about howif location A contains a 2, then either location B is unsafe or.

Moreover, the original rules only pertained to a fixed-size board; inventing a new game played on a 5050 grid would require a whole new set of rules!That is clearly not how we humans conceptualize the game!What we want, when discussing the rules, is a generic way to discussing neighboring locations, so thatwe can have one single rule, saying that if a (generic) location has a zero, then any neighboring location is safe.Thus, we allow the exact details ofneighboring locationto change from game to game as we play on different boards(just as which locations contain pirates changes from game to game).

In a program, you'd probably represent the board as a collection (matrix, list, whatever) of Booleans.In our logic, to correspond to this data structure, we'll introduce binary relations .

By including relations (rather than sticking entirely with propositions), we are leaving the realm of propositional logic;we'll soon reach first-order logic once we also introduce quantifiers corresponding to aspects of program control-flow (loops).
We'll start by adding a way to express whether any two locations are adjacent: a relation nhbr , which will encode the board's geography as follows: nhbr A B and nhbr Z Y are true, while nhbr A D and nhbr M Z are false.

What, exactly, do we mean byrelation? We'll see momentarily , that we can represent nhbr as a set of pairs-of-locations (or equivalently, a function which takes in two locations, and returns either true or false.)

This relation " nhbr " entirely encodes the board's geography. Giving somebody the relation is every bit as good as to showingthem a picture of the board (in some ways, betterthe relation makes it perfectly clear whether two locations which just barely touch at a single point,like B and G , are meant to be considered neighbors.)

We used a binary (two-input) relation to describe neighboring locations.How can we use a relation to capture the notionlocation A is safe?

We'll use a unary (one-input) relation: safe ( A ) is true if and only if (iff) location A is safe.

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After defining relations and discussing their properties, we'll talk about interpreting logic formulas relative to particular relations.

Using relations gives us additional flexibility in modeling our domain, so that our formal logical model more closely corresponds to ourintuition. Relations help separate the WaterWorld domain axioms (code) fromthe data, i.e. , the particular board we're playing on.

Questions & Answers

Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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
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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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hi
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Source:  OpenStax, Intro to logic. OpenStax CNX. Jan 29, 2008 Download for free at http://cnx.org/content/col10154/1.20
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