In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c. Transitivity is a key property of both partial order relations and equivalence relations.
Contents

Formal definition 1

Examples 2

Properties 3

Closure properties 3.1

Other properties 3.2

Properties that require transitivity 3.3

Counting transitive relations 4

See also 5

Sources 6

References 6.1

Bibliography 6.2

External links 7
Formal definition
In terms of set theory, the transitive relation can be defined as:

\forall a,b,c \in X: (aRb \wedge bRc) \Rightarrow aRc
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.
On the other hand, "is the mother of" is not a transitive relation, because if Alice is the mother of Brenda, and Brenda is the mother of Claire, then Alice is not the mother of Claire. What is more, it is antitransitive: Alice can never be the mother of Claire.
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:
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
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]}
See also
Sources
References

^ Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: School of Mathematics  Physics Charles University. p. 1. Lemma 1.1 (iv). Note that this source refers to asymmetric relations as "strictly antisymmetric".

^ Steven R. Finch, "Transitive relations, topologies and partial orders", 2003.

^ Götz Pfeiffer, "Counting Transitive Relations", Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

^ Gunnar Brinkmann and Brendan D. McKay,"Counting unlabelled topologies and transitive relations"
Bibliography

Ralph P. Grimaldi, Discrete and Combinatorial Mathematics, ISBN 0201199122.

Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 9780521762687.
External links

Hazewinkel, Michiel, ed. (2001), "Transitivity",

Transitivity in Action at cuttheknot
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.