Talk:Transitive closure

message: Is there an algorithm, that, given a transitive relation T, will return the smallest relation R such that the transitive closure of R is T?  Is there such a smallest relation?  I'm looking for such an algorithm, and this seems the most logical place to find a link to it.  Luqui 10:55, July 19, 2005 (UTC)
: Yes, and it's called transitive reduction, a concept that Wikipedia is not yet familiar with. Luqui 10:56, July 19, 2005 (UTC)
:: I dispute this.  For finite sets, there are minimal such R, but they are not necessarily unique.  If the relation T is, in addition, antisymmetric or asymmetric, then R is unique.  However, there are minimal reductions of the full relation T on three elements (so that T has 9 pairs) with 3 and with 4 pairs.  In addition, the order relation (<) on the rationals clearly cannot have any minimal reduction. Arthur Rubin 23:23, 15 August 2005 (UTC)
replies: text: There is indeed a smallest one.  To see that, consider that (1) the relation whereby every pair of members of the domain is related, is a transitive relation that is weaker than (i.e. that includes) T, and (2) the set of all such relations is closed under intersection, so the intersection of all such relations is another such relation; that intersection is therefore the smallest one. Michael Hardy 22:44, 19 July 2005 (UTC)
replies: text:  I dispute statement (2) here. Arthur Rubin 23:23, 15 August 2005 (UTC)
replies: text: Sorry -- I read the question wrong.  I thought it was asking whether, given R, there is a smallest T. Michael Hardy 01:40, 16 August 2005 (UTC)

text: The graphviz  tool tred computes a transitive reduction of a graph given as input.  Directed graphs with no cycles (such as partial orders) have a unique transitive reduction. I have found this tool to be helpful for drawing Hasse diagrams. 69.180.211.147 23:23, 17 January 2006 (UTC)

text: The transitive reduction is not unique for cyclic graphs, but the differences between minimal transitive reductions are essentially unimportant: each strongly connected component is represented by a cycle, and if we contract each of these cycles to a vertex we will always get the same graph. We can attain a canonical form by numbering the vertices, promising that all edges not in the cycle go in and out from the lowest-numbered vertex, and choose the cycle to pass through the vertices in increasing number order (and back around from the largest to the smallest). Deco 23:43, 17 January 2006 (UTC)

text: So it's clear that there should be an article on transitive reduction.  Should we move this bit of discussion there? 69.180.211.147 01:10, 18 January 2006 (UTC)

text: OK, i made my first article: transitive reduction Lyonsam 02:03, 18 January 2006 (UTC)
replies: text: Good job! I'll add a bunch of stuff to it. Deco 03:12, 18 January 2006 (UTC)
thread_title:  Minimal inverse 

message: Would be really nice if this page could have a slightly less technical explanation too. The term "transitive closure" is sometimes used in less formal contexts, it would be nice to know what it meant! —The preceding unsigned comment was added by 80.111.233.163 (talk • contribs) 18:35, 04 September 2005  (UTC)
:What do you have in mind?  The first sentence seems to be a good non-technical summary. Arthur Rubin 18:35, 7 September 2005 (UTC)
replies: text: Right. I came here after my textbook did a lousy job at explaining it, and this is even worse.  Maybe it could do with some explanations that don't involve so many symbols. --BennyD 19:06, 16 December 2006 (UTC)
thread_title:  *blink* 

message: What set theory allows, for example, humans and airports to be members of sets?  What a silly thing to say.  —Preceding unsigned comment added by 24.61.183.197 (talk • contribs) 02:02, April 19, 2007
Furthermore, since when is being an ancestor imply being a parent of said descendant? And certain a flight with a connection does not imply that it is a direct flight. These examples should be removed entirely MountainGoat8 (talk) 23:38, 19 October 2009 (UTC)
replies: text: Being an ancestor does not, per the description, imply being a parent.  You may be a parent of a parent and still fit the definition, an so on up to an arbitrary number of "parent-of"s.  Can you point out where the description confused you?  Perhaps distinction between R and the transitive closure of R was not strong enough? 129.93.158.25 (talk) 23:31, 24 October 2009 (UTC)
thread_title: nonmathematical

message: It would be helpful to relate this definition to the definition of closure, which uses the example of subtraction over the real numbers (applying it to any two elements in that set yields a result that is also in that set.)
For example, if we start from a specific city x and apply the given transitive closure to yield the set of cities that can be reached from x, this set is closed in the sense that applying the same relation to any of those cities yields the same set again:  you can't get to any other city but those in the set.  But it's not quite the same, because the relation for cities operates on a single city, and yields a set, which are different kinds of things.  In contrast, subtraction operates on, and yields, single elements of the set of real numbers.
Is the resolution to this to start with a set of cities (instead of a city), and define the domain as the set of sets of cities?
203.217.66.200 (talk) 07:13, 13 May 2010 (UTC)
replies: text: The two meanings are rather distinct; note the lack of "transitive" when one speaks of binary operations. Any function is also a relation. If x + y is closed to a subset, say Z of integers, then +(x, y, z), where z = x + y, is a relation on Z. But it's a ternary rather than a binary one, so there's not much else that can be said. I don't think one can define a meaningful notion of transitive closure for n-ary, n>2 relations or equivalently for hypergraphs, even though the latter can be embedded in Levi graphs. If you apply the transitive closure notion to the Levi graph of addition, you simply say that 1+3 = 4 = 2+2 for instance, because there's an edge from (1,3) to 4 and another from (2, 2) to 4. Tijfo098 (talk) 10:15, 11 November 2012 (UTC)
thread_title:  How is this definition related to the definition of "closure"? 

message: The intro currently reads '...is the transitive relation R+ on set X such that R+ contains R and R+ is minimal'.
Before , it said '...is the smallest transitive relation on X that contains R'.
I find the old version much better. The word 'minimal' is confusing - to me it sounds like being minimal is an absolute property of a relation, although it is just relative to other transitive relations that contain R.
If no-one objects, I'll change it back soon.
Chrisahn (talk) 21:58, 15 May 2012 (UTC)
thread_title:  'Minimal' vs 'smallest' 

message: The opening paragraph contains the sentence "In set theory, binary relations are represented as sets, hence "smallest" pertains to set cardinality."
This is just complete nonsense, in general there is no "smallest" subset of a set in terms of cardinality.  In particular, for any infinite set, adding a single element will lead to a set of equal cardinality.
Therefore, this wrong sentence should either be removed or replaced by the correct statement that the term "minimal" here is taken with respect to the partial ordering of "being subset of".  Since being transitive is closed under intersection, the minimal element is actually a minimum and thus well-defined.
84.156.209.58 (talk) 11:00, 20 March 2021 (UTC)
replies: text: - I agree with you in that "cardinality" shouldn't appear here. However, I hesitated to put your explanation about uniqueness in the very first sentence of the lead. The same applies for the explanation about relations being (implemented as) sets. Therefore, I used "material implication" (with a slightly WP:EASTEREGGy link) as ordering relation. - Jochen Burghardt (talk) 16:59, 20 March 2021 (UTC)
replies: text: I think that is totally confusing as an explanation of what smallest means. What is implying what? How does material implication define an ordering on things? The simpler explanation is that it is the unique minimal relation. —David Eppstein (talk) 21:00, 20 March 2021 (UTC)
thread_title:  Smallest with respect to cardinality 

message: Is "power set of" missing in "Furthermore, there exists at least one transitive relation containing R, namely the trivial one: X × X." ? --Rainald62 (talk) 20:28, 22 July 2023 (UTC)
replies: text: No, it isn't. If X = {1,2}, then X×X = {(1,1),(1,2),(2,1),(2,2)} is the relation that relates every member of X with every member of X. The power set of X×X is the set of all 16 subsets of X×X, viz. { {},  {(1,1)}, {(1,2)}, {(2,1)}, {(2,2)},  {(1,1),(1,2)}, {(1,1),(2,1)}, {(1,1),(2,2)}, {(1,2),(2,1)}, {(1,2),(2,2)}, {(2,1),(2,2)},  {(1,1),(1,2),(2,1)}, {(1,1),(1,2),(2,2)}, {(1,1),(2,1),(2,2)}, {(1,2),(2,1),(2,2)},  {(1,1),(1,2),(2,1),(2,2)} }. This is a set of relations, but not a relation. - Jochen Burghardt (talk) 10:59, 23 July 2023 (UTC)
thread_title:  Existence