In this case we are studying transitivity in relation to Computer networking. I suppose the biggest use of this concept might be in relation to TCP/IP or Transfer control protocol/ Internet Protocol which allows all devices (no matter the computer languages used in their operating systems to be able to communicate with each other on some basic level to allow individual devices like cell phone (smartphones) and IPads and tablets, and laptops and desktops and Servers and mainframes to all communicate in one or more ways ongoing throughout the world.
It is sort of like how airline pilots all speak English now all over the world. This is one of the prerequisites for being an airline pilot. The reason for this is simple: there has to be one language they all speak or in an emergency people are going to die or get maimed. The language could have been any language but I think likely after World War II for sure this became the norm because the U.S. was the only large country with infrastructure mostly not destroyed by the war on earth left.
In a similar sense in dealing with a worldwide Internet there must be one language that translates all the others into a usable form which is TCP/IP. So, transivity is making sure communication can go through multiple communication languages in regards to a multiplicity of devices and reach a common meaning or somewhat common meaning between them all.
One of the algebraic formulas for figuring all this out is:
For example, "is greater than," "is at least as great as," and "is equal to" (equality) are transitive relations:- whenever A > B and B > C, then also A > C
- whenever A ≥ B and B ≥ C, then also A ≥ C
- whenever A = B and B = C, then also A = C
Transitivity
Transitive relation
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Contents
Formal definition
In terms of set theory, the transitive relation can be defined as:
Examples
For example, "is greater than," "is at least as great as," and "is equal to" (equality) are transitive relations:- whenever A > B and B > C, then also A > C
- whenever A ≥ B and B ≥ C, then also A ≥ C
- whenever A = B and B = C, then also A = C.
Then again, in biology we often need to consider motherhood over an arbitrary number of generations: the relation "is a matrilinear ancestor of". This is a transitive relation. More precisely, it is the transitive closure of the relation "is the mother of".
More examples of transitive relations:
- "is a subset of" (set inclusion)
- "divides" (divisibility)
- "implies" (implication)
Properties
Closure properties
The converse of a transitive relation is always transitive: e.g. knowing that "is a subset of" is transitive and "is a superset of" is its converse, we can conclude that the latter is transitive as well.The intersection of two transitive relations is always transitive: knowing that "was born before" and "has the same first name as" are transitive, we can conclude that "was born before and also has the same first name as" is also transitive.
The union of two transitive relations is not always transitive. For instance "was born before or has the same first name as" is not generally a transitive relation.
The complement of a transitive relation is not always transitive. For instance, while "equal to" is transitive, "not equal to" is only transitive on sets with at most one element.
Other properties
A transitive relation is asymmetric if and only if it is irreflexive.[1]Properties that require transitivity
- Preorder – a reflexive transitive relation
- partial order – an antisymmetric preorder
- Total preorder – a total preorder
- Equivalence relation – a symmetric preorder
- Strict weak ordering – a strict partial order in which incomparability is an equivalence relation
- Total ordering – a total, antisymmetric transitive relation
Counting transitive relations
No general formula that counts the number of transitive relations on a finite set (sequence A006905 in OEIS) is known.[2] However, there is a formula for finding the number of relations that are simultaneously reflexive, symmetric, and transitive – in other words, equivalence relations – (sequence A000110 in OEIS), those that are symmetric and transitive, those that are symmetric, transitive, and antisymmetric, and those that are total, transitive, and antisymmetric. Pfeiffer[3] has made some progress in this direction, expressing relations with combinations of these properties in terms of each other, but still calculating any one is difficult. See also.[4]| Number of n-element binary relations of different types | ||||||||
|---|---|---|---|---|---|---|---|---|
| n | all | transitive | reflexive | preorder | partial order | total preorder | total order | equivalence relation |
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 16 | 13 | 4 | 4 | 3 | 3 | 2 | 2 |
| 3 | 512 | 171 | 64 | 29 | 19 | 13 | 6 | 5 |
| 4 | 65536 | 3994 | 4096 | 355 | 219 | 75 | 24 | 15 |
| OEIS | A002416 | A006905 | A053763 | A000798 | A001035 | A000670 | A000142 | A000110 |
See also
Sources
References
- Gunnar Brinkmann and Brendan D. McKay,"Counting unlabelled topologies and transitive relations"
Bibliography
- Ralph P. Grimaldi, Discrete and Combinatorial Mathematics, ISBN 0-201-19912-2.
- Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.
External links
- Hazewinkel, Michiel, ed. (2001), "Transitivity", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Transitivity in Action at cut-the-knot
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