# 3.1 Properties of relations  (Page 2/2)

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The next two exercises aren't meant to be difficult, but rather to illustrate that, while we've sketched these twoapproaches and suggested they are equivalent, we still need an exact definition.

For the indicator function $f(x, y)=\begin{cases}\mbox{true} & \text{if y=x^{2}}\\ \mbox{false} & \text{otherwise}\end{cases}$ on the domain of (pairs of) natural numbers, write down the set-of-pairs representationfor the corresponding binary relation. It's insightful to give the answer both by listing the elements,possibly with ellipses, and also by using set-builder notation.

In general , for a binary indicator function $f$ , what, exactly, is the corresponding set?

$\{\left(0,0\right), \left(1,1\right), \left(2,4\right), \left(3,9\right), \text{…}, \left(i,i^{2}\right), \text{…}\}$ In set-builder notation, this is $\{\left(x,y\right)\colon y=x^{2}\}$

In general, for an indicator function $f$ , the correspondingset would be $\{\left(x,y\right)\colon f(x, y)\}$ (Note that we don't need to write

$f(x, y)=\mbox{true}$
; as computer scientists comfortable with Booleansas values, we see this is redundant.)

For the relation $\mathrm{hasPirate}=\{K, T, R, U, E\}$ on the set of (individual) WaterWorld locations, write down the indicator-function representationfor the corresponding unary relation. In general , how would you write down this translation?

$f_{\mathrm{hasPirate}}(x)=\begin{cases}\mbox{true} & \text{if x=K}\\ \mbox{true} & \text{if x=T}\\ \mbox{true} & \text{if x=R}\\ \mbox{true} & \text{if x=U}\\ \mbox{true} & \text{if x=E}\\ \mbox{false} & \text{otherwise}\end{cases}$ .

In general, for a (unary) relation $R$ , $f_{R}(x)=\begin{cases}\mbox{true} & \text{if x\in R}\\ \mbox{false} & \text{if x\notin R}\end{cases}$ .

Since these two formulations of a relation, sets and indicator functions,are so close, we'll often switch between them (a very slight abuse of terminology).

Think about when you write a program that uses the abstract data type Set . Its main operation is elementOf . When might you use an explicit enumeration to encode a set,and when an indicator function? Which would you use for the set of WaterWorld locations?Which for the set of prime numbers?

## Functions as relations

Some binary relations have a special property: each element of the domain occurs as the first itemin exactly one tuple. For example, $\mathrm{isPlanet}=\{\left(\mathrm{Earth},\mbox{true}\right), \left(\mathrm{Venus},\mbox{true}\right), \left(\mathrm{Sol},\mbox{false}\right), \left(\mathrm{Ceres},\mbox{false}\right), \left(\mathrm{Mars},\mbox{true}\right)\}$ is actually a (unary) function. On the other hand, $\mathrm{isTheSquareOf}=\{\left(0,0\right), \left(1,1\right), \left(1,-1\right), \left(4,2\right), \left(4,-2\right), \left(9,3\right), \left(9,-3\right), \text{…}\}$ is not a function, for two reasons. First, some numbers occur as the first element of multiple pairs. Second, some numbers, like $3$ , occurs as the first element of no pairs.

We can generalize this to relations of higher arity, also. This is explored in this exercise and this one .

## Binary relations

One subclass of relations are common enough to merit some special discussion: binary relations. These are relations on pairs, like $\mathrm{nhbr}$ .

## Binary relation notation

Although we introduced relations with prefix notation, e.g., $<(i, j)$ , we'll use the more common infix notation, $i< j$ , for well-known arithmetic binary relations.

## Binary relations as graphs

In fact, binary relations are common enough that sometimes people use some entirely new vocabulary:A domain with a binary relation can be called vertices with edges between them. Together this is known as a graph . We won't stress these terms right now,as we're not studying graph theory.

Binary relations (graphs) can be depicted visually, by drawing the domain elements (vertices) as dots,and drawing arrows (edges) between related elements.

A binary relation with a whole website devoted to it is

has starred in a movie with
. We'll call this relation $\mathrm{hasStarredWith}$ over the domain of actors. Some sample points in this relation:
• $\mathrm{hasStarredWith}(\mathrm{Ewan McGregor}, \mathrm{Cameron Diaz})$ , as witnessed by the movie A Life Less Ordinary , 1997.
• $\mathrm{hasStarredWith}(\mathrm{Cameron Diaz}, \mathrm{John Cusack})$ , as witnessed by the movie Being John Malkovich , 1999.
You can think of each actor being a
location
, and two actors being
to each other if they have ever starred in a movie together;two of these locations, even if not adjacent might have a multi-step path between them.(There is also a shorter path; can you think of it?The (in)famous Kevin Bacon game asks to find a shortest path from one location to thelocation Kevin Bacon. Make a guess, as to the longest shortest path leading from(some obscure) location to Kevin Bacon.)

Some other graphs:

• Vertices can be tasks, with edges meaning dependencies of what must be done first.
• In parallel processing, Vertices can be lines of code;there is an edge between two lines if they involve common variables.Finding subsets of vertices with no lines between them represent sets of instructions that can beexecuted in parallel (and thus assigned to different processors.)
seek to transform one word to another by changing one letter at a time, while always remaining a word.For example, a ladder leading from WHITE to SPINE in three steps is:
• WHITE
• WHINE
• SHINE
• SPINE
If a solution to such a puzzle corresponds to a path, what do vertices represent?What are edges? Do you think there is a path from any 5-letter word to another?

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absolutely yes
Daniel
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it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
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what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
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Cesar
I'm interested in nanotube
Uday
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preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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