# 0.1 Discrete structures relation  (Page 11/13)

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The poset of the set of positive real numbers with the less-than-or-equal-to relation is not a well order, because the set itself does not have any least element (0 is not in the set).

A digraph of a binary relation on a set can be simplified if the relation is a partial order. Hasse diagrams defined as follows are such graphs.

Definition(Hasse diagram): A Hasse diagram is a graph for a poset which does not have loops and arcs implied by the transitivity. Further, it is drawn so that all arcs point upward eliminating arrowheads.

To obtain the Hassse diagram of a poset, first remove the loops, then remove arcs<a, b>if and only if there is an element c that<a, c>and<c, b>exist in the given relation.

Example 10: For the relation {<a, a>,<a, b>,<a, c>,<b, b>,<b, c>,<c, c>} on set {a, b,c}, the Hasse diagram has the arcs {<a, b>,<b, c>} as shown in Figure 10.

## Topological sorting

The elements in a finite poset can be ordered linearly in a number of ways while preserving the partial order. For example {∅, {1}, {2}, {1, 2}} with the partial order ⊆, can be ordered linearly as ∅, {1}, {2}, {1, 2}, or ∅, {2}, {1}, {1, 2}. In these orders a set appears before (to the left of) another set if it is a subset of the other. In real life, tasks for manufacturing goods in general can be partially ordered based on the prerequisite relation, that is certain tasks must be completed before certain other tasks can be started. For example the arms of a chair must be carved before the chair is assembled. Scheduling those tasks is essentially the same as arranging them with a linear order (ignoring here some possible concurrent processing for simplicity's sake).

The topological sorting is a procedure to find from a partial order on a finite set a linear order that does not violate the partial order. It is based on the fact that a finite poset has at least one minimal element. The basic idea of the topological sorting is to first remove a minimal element from the given poset, and then repeat that for the resulting set until no more elements are left. The order of removal of the minimal elements gives a linear order. The following algorithm formally describes the topological sorting.

Algorithm Topological Sort

Input: A finite poset<A, R>.

Output: A sequence of the elements of A preserving the order R.

integer i;

i := 1;

while ( A ≠∅) {

pick a minimal element b from A;

A := A - {b};

i := i + 1;

output b

} Example: Let A = {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} with the partial order ⊆. This given A has three minimal elements {1}, {2}, and {3}.

Select {2} and remove it from A. Let A denote the resultant set i.e. A := A - {2}. The new A has two minimal elements {1}, and {3}.

Select {1} and remove it from A. Denote by A the resultant set, that is A = {{3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}.

This new A has two minimal elements {3} and {1, 2}.

Select {1, 2} and remove it from A.

Proceeding in like manner, we can obtain the following linear order: {{2}, {1}, {1, 2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}.

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Joseph
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where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
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da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
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Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
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narayan
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ya I also want to know the raman spectra
Bhagvanji
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Damian
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
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Rafiq
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What is meant by 'nano scale'?
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LITNING
scanning tunneling microscope
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Rafiq
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Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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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
how did you get the value of 2000N.What calculations are needed to arrive at it
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