# 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?

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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