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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

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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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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