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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

are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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