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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 is a wave?
DAVID Reply
show that coefficient of friction of solid block inclined at an angle is equivalent to trignometric tangent of angle
DAVID
thanks for that definition.
Dodou Reply
Hi everyone please can dere be motion without force?
Lafon
no...
Enyia
Thanks
Lafon
hi
Omomaro
whats is schrodinger equation
Omomaro
l went spiral spring
Xalat
what is position?
Adhar Reply
position is simply where you are or where you were
Shii
position is the location of an object with respect to a two or three dimensional axes or space.
Bamidele
Can dere be motion without force?
Lafon
what is the law of homogeinity?
auson Reply
two electric lines of force never interested each other. why?
Sujit Reply
proof that for BBC lattice structure 4r\root 5 and find Apf for the BBC structure
Eric Reply
what is physics?
Abdulaziz Reply
physics is deine as the specific measrument of of volume, area,nd distances...
Olakojo
if a string of 2m is suspended an an extended 3m elasticity is been applied.... is hooks law obeyed?
Enyia
if a string of 2m is suspended an an extended 3m elasticity is been applied.... is hooks law obeyed?
Enyia
yes
Alex
proof that for a BBC lattice structure a= 4r/ root 5 find the APF for the BBC structure
Eric
if a string of 2m is suspended an an extended 3m elasticity is been applied.... is hooks law obeyed?
Enyia Reply
tell me conceptual quetions of mechanics
Syeda Reply
I want to solve a physical question
ahmed
ok
PUBG
a displacement vector has a magnitude of 1.62km and point due north . another displacement vector B has a magnitude of 2.48 km and points due east.determine the magnitude and direction of (a) a+ b and (b) a_ b
Kou Reply
quantum
George
a+b=2.9
SUNJO
a+b
Yekeen
use Pythogorous
Dhritwan
A student opens a 12kgs door by applying a constant force of 40N at a perpendicular distance of 0.9m from the hinges. if the door is 2.0m high and 1.0m wide determine the magnitude of the angular acceleration of the door. ( assume that the door rotates freely on its hinges.) please assist me to d
Mike
what is conditions met to produce shm
Enocy Reply
what is shm
Manzoor
shm?
Grant
Why is Maxwell saying that light is an electromagnetic wave?
Bong
1st condition; It(th e BBC's system) must have some inertia which will enable it to possess Kinetic energy 2. must be able to store potential energy
Calleb
I meant "the system" not the BBC'S....."
Calleb
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Manzoor
kindly tell us the name of your university
Manzoor
GUlam Ishaq Khan INSTITUTE of engineering science
ali
Department of Environment Ionian University Zante Greece
why light wave travel faster than sounds
ALI Reply
Why light travel faster than sounds?
ALI
Light travel faster than sound because it does not need any medium to travel through.
alhassan
when an aeroplane flies....why it does not fall on the earth?
Frazali
As an aeroplane moves, it hits a wind,we have the wind flowing at the upper and lower zone of the aeroplane, the one that is moving on the upper zone moves at a greater speed than that of the lower zone, this creates a low pressure on the upper zone and a greater pressure at the lower zone.
Kipkoech
which thing of aeroplane moves it upward?
Frazali
good question
Manzoor
about force
Barataa
am pleased to join the group
Nesru
yea
caleb
It a privilege to be here
olajire
hi
Awode
hello
Manzoor
Light speed is more than sound speed. C=3×10*8m/s V=320-340 m/s
siva
A body of mass 2kg slides down a rough plane inclined to horizontal at 30degrees. find the energy that is wasted as a result of friction if the co-efficient of kinetic f
official Reply
ten applications of Newton's second law of motion
Alale Reply
Calculate the volume at S.T.P of a gas whose volume at -5° and 746 mmHg
Mlungisi 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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