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Definition (transitive relation): A relation R on a set A is called transitive if and only if for any a, b, and c in A, whenever<a, b>∈R , and<b, c>∈R ,<a, c>∈R .

Example 8: The relation ≤ on the set of integers {1, 2, 3} is transitive, because for<1, 2>and<2, 3>in ≤,<1, 3>is also in ≤, for<1, 1>and<1, 2>in ≤,<1, 2>is also in ≤, and similarly for the others. As a matter of fact ≤ on any set of numbers is also transitive. Similarly ≥ and = on any set of numbers are transitive.

Figure 5 show the digraph of relations with different properties.

(a) is reflexive, antisymmetric, symmetric and transitive, but not irreflexive.

(b) is neither reflexive nor irreflexive, and it is antisymmetric, symmetric and transitive.

(c) is irreflexive but has none of the other four properties.

(d) is irreflexive, and symmetric, but none of the other three.

(e) is irreflexive, antisymmetric and transitive but neither reflexive nor symmetric.

Operations on binary relations

Set operations

A relation is a set. It is a set of ordered pairs if it is a binary relation, and it is a set of ordered n-tuples if it is an n-ary relation. Thus all the set operations apply to relations such as ∪, ∩, and complementing.

For example, the union of the "less than" and "equality" relations on the set of integers is the "less than or equal to" relation on the set of integers. The intersection of the "less than" and "less than or equal to" relations on the set of integers is the "less than" relation on the same set. The complement of the "less than" relation on the set of integers is the "greater than or equal to" relation on the same set.

Composite relations

If the elements of a set A are related to those of a set B, and those of B are in turn related to the elements of a set C, then one can expect a relation between A and C. For example, if Tom is my father (parent-child relation) and Sarah is a sister of Tom (sister relation), then Sarah is my aunt (aunt-nephew/niece relation). Composite relations give that kind of relations.

Definition(composite relation): Let R1 be a binary relation from a set A to a set B, R2 a binary relation from B to a set C. Then the composite relation from A to C denoted by R1R2 (also denoted by R1 ∘ R2 is defined as

R1R2 = {<a, c>| a ∈A ⋀c ∈C ⋀∃b [b ∈B ⋀<a, b>∈R1 ⋀<b, c>∈R2 ] } .

In English, this means that an element a in A is related to an element c in C if there is an element b in B such that a is related to b by R1 and b is related to c by R2. Thus R1R2 is a relation from A to C via B in a sense. If R1 is a parent-child relation and R2 is a sister relation, then R1R2 is an aunt-nephew/niece relation.

Example 1: Let A = {a1 , a2} , B = {b1 , b2 , b3} , and C = {c1 , c2} . Also let R1 = {<a1 , b1>,<a1 , b2>,<a2 , b3>} , and R2 = {<b1 , c1>,<b2 , c1>,<b2 , c2>,<b3 , c1>} . Then R1R2 = {<a1 , c1>,<a1 , c2>,<a2 , c1>} .

This is illustrated in Figure 6. The dashed lines in the figure of R1R2 indicate the ordered pairs in R1R2, and dotted lines show ordered pairs that produce the dashed lines. (The lines in the left figure are all supposed to be solid lines.)

Questions & Answers

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.
Damian
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Discrete structures. OpenStax CNX. Jan 23, 2008 Download for free at http://cnx.org/content/col10513/1.1
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