# c++ program to find transitive relation

How can I make sure I'll actually get it? Finally, it checks that no entry in the latter matrix exceeds that in the adjacency matrix. Code Golf Stack Exchange is a question and answer site for programming puzzle enthusiasts and code golfers. Physicists adding 3 decimals to the fine structure constant is a big accomplishment. Is there any way that a creature could "telepathically" communicate with other members of it's own species? Taking the boring O(n^3) approach. If the input list contains {a,b} and {b,c} but not {a,c} for some a, b, c, replaces it with 0. However, please refrain from exploiting obvious loopholes. Writing a Simple Program in … ... Well, for finding transitive closure, we don't need to worry about the weighted edges and we only need to see if there is a path from a starting vertex i to an ending vertex j. Und zwar soll ich in meinen Uebungsaufgaben verschiedene Relationen finden. Ich hatte an x+x = gerade Zahl gedacht. Program on Relations. You will be given a list of pairs of integers in any reasonable format. Wow, Clojure for loops are cool :D This checks that the for loop does not generate a falsy value, which occurs if [a d] is not found from the input set. Why? To start viewing messages, select the forum that you want to visit from the selection below. * permissions. Usage example: f [(1,2), (2,4), (6,5), (1,4)]-> True. You may have to register or Login before you can post: click the register link above to proceed. Reachable mean that there is a path from vertex i to j. Eine davon soll reflexiv, symmetrisch und transitiv sein. For the same FOV and f-stop, will total luminous flux increase linearly with sensor area? E.g. So a ~* b means either a ~ b or there is a c so that a ~* c and c ~ b. @MartinEnder I think you misinterpreted "ordered pairs". For instance "was born before or has the same first name as" is not generally a transitive relation. For instance, [(1, 2), (5, 1), (-9, 12), (0, 0), (3, 2)] is a relation. Then again, in biology we often need to … Output: a truthy value for a transitive relation, falsy otherwise. The relation is not transitive if we can find (A,B) and (B,C) such that (A,C) doesn't hold. We will also see the application of Floyd Warshall in determining the transitive closure of a given graph. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. If you think a specification is unclear or underspecified, comment on the question instead. a = 1. b = 2. c = 3. You can always include a readable version of the code in addition to the competitive one. Year: May 2015. mumbai university discrete structures • 5.8k views. Then the transitive closure of R is the connectivity relation R1.We will now try to prove this C++ Program to Find the Transitive Closure of a Given Graph G. C++ Server Side Programming Programming. For instance, knowing that "is a subset of" is transitive and "is a superset of" is its inverse, one can conclude that the latter is transitive as well. Relation & Function - In many naturally occurring phenomena, two variables may be linked by some type of relationship. a R b, b R c -----> a R c. Example : Let A = { 1, 2, 3 } and R be a relation defined on set A as "is less than" and R = {(1, 2), (2, 3), (1, 3)} Verify R is transitive. To learn more, including how to choose permissions, see Permissions. That brings us to the concept of relations. The union of two transitive relations is not always transitive. rev 2020.12.3.38123, The best answers are voted up and rise to the top, Code Golf Stack Exchange is a site for recreational programming competitions, not general programming questions. Given a list of pairs of integers, determine if a relation is transitive or not. Due: Mon, Nov.10, 2014. …Avoid asking for help, clarification or responding to other answers (use comments instead). Waere das richtig? Marks: 8 Marks. Find transitive closure using Warshall's Algorithm. Then, we have (a, b) = (1, 2) -----> 1 is less than 2 (b, c) = (2, 3) -----> 2 is less than 3 Combine the results with logical and. we need to find until . Both these solutions are unnamed functions taking a list of ordered pairs as input and returning True or False. Why does this movie say a witness can't present a jury with testimony which would assist in making a determination of guilt or innocence? In case r is an equivalence relation, you are to find and output the equivalence classes. (a, b), (c, d) present such that b = c. [(5, 9), (9, 54), (0, 0)] is not transitive, because it contains (5, 9) and (9, 54), but not (5, 54). Mumbai University > Computer Engineering > Sem 3 > Discrete Structures. What is more, it is antitransitive: Alice can neverbe the mother of Claire. C Program to implement Warshall’s Algorithm Levels of difficulty: medium / perform operation: Algorithm Implementation Warshall’s algorithm enables to compute the transitive … Falsy is a matrix that contains at least one zero. So the result is the set of ordered pairs that needs to be in the input for it to be transitive; SubsetQ[#,...] detects that property. What is the physical effect of sifting dry ingredients for a cake? …Be sure to follow the challenge specification. Answers abusing any of the standard loopholes are considered invalid. The complement of a transitive relation is not always transitive. Beispiele für transitive Relationen sind die Gleich- und die Kleiner-Relationen auf den reellen Zahlen, denn für drei reelle Zahlen , und mit = und = gilt immer auch =, und aus < und < folgt <. Add single unicode (euro symbol) character to font under Xe(La)TeX. Transitive: Relation R is transitive because whenever (a, b) and (b, c) belongs to R, (a, c) also belongs to R. Example: (3, 1) ∈ R and (1, 3) ∈ R (3, 3) ∈ R. So, as R is reflexive, symmetric and transitive, hence, R is an Equivalence Relation. Why does a firm make profit in a perfect competition market. Is there an "internet anywhere" device I can bring with me to visit the developing world? In contrast, a func Saved 2 bytes thanks to an idea by @Cyoce. This reach-ability matrix is called transitive closure of a graph. For instance, answers to code-golf challenges should attempt to be as short as possible. Is there a general solution to the problem of "sudden unexpected bursts of errors" in software? Following this channel's introductory video to transitive relations, this video goes through an example of how to determine if a relation is transitive. Data structures using C, Here we solve the Warshall’s algorithm using C Programming Language. @isaacg that's what I meant. @Lynn Check out the Prolog answer, then ;-), You might be able to shorten the second solution by making an abbreviation for, @Cyoce Indeed, you save 3 bytes each time by writing. …Include a short header which indicates the language(s) of your code and its score, as defined by the challenge. Note1: If R 1 and R 2 are equivalence relation then R 1 ∩ R 2 is also an equivalence relation. One graph is given, we have to find a vertex v which is reachable from … Truthy output is a matrix formed by ones. Tips to stay focused and finish your hobby project, Podcast 292: Goodbye to Flash, we’ll see you in Rust, MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC…, Language of the month for November 2020: Forth, Find maximal matching in divisibility relation, Equivalence classes on the Transitive Closure of a Reflexive Relation. Adventure cards and Feather, the Redeemed? C Program to Swap two Numbers; Program to check if a given year is leap year; C Program to print Floyd’s triangle; Program to find area of a circle ; Program to find area of a triangle; Program for factorial of a number; Factorial of a large number; Factorial of Large numbers using Logarithmic identity; Compute n! Let V [ i , j ] be optimal value of such instance. Then we return truthy from the inner predicate (falsey from the whole program) if [A:C] isn't there. (e,e)} On {a,b,c,d,e} Do Not Use Any Other Method. Those are then operated upon by the function If[#2==#3,{#,#4},Nothing]&@@@, which has a cool property: if the middle two elements are equal, it returns the ordered pair consisting of the first and last numbers; otherwise it returns Nothing, a special Mathematica token that automatically disappears from lists. My manager (with a history of reneging on bonuses) is offering a future bonus to make me stay. [(1, 2), (2, 4), (6, 5), (1, 4)] is transitive, because it contains (1, 2) and (2, 4), but (1, 4) as well. You should have a test case that is only transitive because the pairs are ordered. Composition of Relations. Reachable mean that there is a path from vertex i to j. (Outer)loop over all pairs (a,b) and (inner)loop over the same pairs, now called (c,d) and every time when b==c check if (a,d)is also an existent pair. In other words, my test case is only truthy because the relation isn't implicitly symmetric. Is the energy of an orbital dependent on temperature? The function of those two indices is Last@i!=#&@@j||#~MemberQ~{#&@@i,Last@j}, which translates to "either the second element of i and the first element of j don't match, or else the input contains the ordered pair consisting of the first element of i and the last element of j". Requires its input to be a container of pair. Eine transitive Relation ist in der Mathematik eine zweistellige Relation auf einer Menge, die die Eigenschaft hat, dass für drei Elemente , , dieser Menge aus und stets folgt. This post covers in detail understanding of allthese With this behaviour applications can request the least privileged permissions they need, rather than rely on the set of Directory. Warshall’s algorithm enables to compute the transitive closure of the adjacency matrix of any digraph. The intersection of two transitive relations is always transitive. Die Frage ist, ob dass jetzt schon eine Äquivalenzrelation ist. What does it mean to “key into” something. Consider a relation. Solution : From the given set A, let. For instance, while "equal to" is transitive, "not equal to" is only transitive on sets with at most one element. There were four previous 69-byte formulations: In other words, if the input contains pairs [A:B] and [B:C], we can permute the input to put [A:B] and [B:C] at the start, delete all other elements, and produce a list [A:B:B:C]. Results 1 to 4 of 4 Thread: Transitive relation. You may assume that the input will consist of at least one pair, and that the pairs are unique. (6.a), (b, D), (c,d). Then Show That R1 R2 Is Irreflexive! Schon mal vielen Dank für eure Antworten relation; eigenschaften; transitiv; reflexiv; symmetrie ; Gefragt 28 Okt 2017 von Mathedude99. Question: C++ PROGRAM FOR MATRIX RELATIONS (reflexivity, Transitivity, Symmetry, Equivalance Classes) Need Help Completing The Functions, Thanks /* Reads In A Matrix From A Binary File And Determines RST And EC's. This is an anonymous block (function) which takes the elements as a two-level array, but the test suite does string manipulation to put the input into a suitable format first. 2 TRANSITIVE CLOSURE 2 Transitive Closure A relation R is said to be transitive if for every (a;b) 2 R and (b;c) 2 R there is a (a;c) 2 R.A transitive closure of a relation R is the smallest transitive relation containing R. Suppose that R is a relation deﬂned on a set A and that R is not transitive. For details, see … You are to write one program to determine whether or not r is reflexive, symmetric, transitive, antisymmetric, an equivalence relation. Question: Q4a) Find The Transitive Closure Using Warshall's Algorithm Of The Relation Given By {(a, E). Difference between reflexive and identity relation. What do I do to get my nine-year old boy off books with pictures and onto books with text content? factorial. As unnamed lambda returning via reference parameter. This input has to be a set of two-element vectors: If input must be list-like then (%[a d]) has to be replaced by ((set %)[a d]) for extra 6 bytes. Program to print the Union And Intersection of a P... Count Number of Digits in N! Does the input have to be a list-like format, or can it be an adjacency--matrix-like format? A relation R on A is said to be a transitive relation if and only if, (a,b) $\in$ R and (b,c) $\in$ R $\Rightarrow$ (a,c) $\in$ R for all a,b,c $\in$ A. that means aRb and bRc $\Rightarrow$ aRc for all a,b,c $\in$ A. Update: Instead of using :when I'll just check that for all pairs of [a b] [c d] either b != c or [a d] is found from the input set. This creates a 2D array of booleans, which And@@And@@@ flattens into a single boolean. (d, A),(d.c), (e,a), (e,b), (e,c). Are the natural weapon attacks of a druid in Wild Shape magical? Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Online test suite. I don't think it means the pairs in an order - I think it means each pair has an order, first then second. Since, we stop the process. Table[...,{i,#},{j,#}] creates a 2D array indexed by i and j, which are taken directly from the input (hence are both ordered pairs). Transitive closure, – Equivalence Relations : Let be a relation on set . If a directed graph is given, determine if a vertex j is reachable from another vertex i for all vertex pairs (i, j) in the given graph. Challenges must have, Code Golf Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. If a relation is Reflexive symmetric and transitive then it is called equivalence relation. For a relation R in set AReflexiveRelation is reflexiveIf (a, a) ∈ R for every a ∈ ASymmetricRelation is symmetric,If (a, b) ∈ R, then (b, a) ∈ RTransitiveRelation is transitive,If (a, b) ∈ R & (b, c) ∈ R, then (a, c) ∈ RIf relation is reflexive, symmetric and transitive,it is anequivalence relation int i=0; //flag for local transitivity for(auto c:m) //search for a third element i+=a.first==c.first&&b.second==c.second; r*=i>0; //multiply with flag>0, resulting in 0 forever if one was not found } } ; int main(){ std::vector> m={ {1, 2}, {2, 4}, {6, 5}, {1, 4} }; int r; f(m,r); std::cout << r << std::endl; m.emplace_back(3,6); f(m,r); std::cout << r << std::endl; m.emplace_back(3,5); f(m,r); … For the transitive closure, we need to find . Transitive Relation Let A be any set. Visual Basic 6.0 Programming; Transitive relation; If this is your first visit, be sure to check out the FAQ by clicking the link above. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. [(7, 8), (9, 10), (15, -5)] is transitive, because there aren't any two pairs It only takes a minute to sign up. Equivalence relation. -24 bytes thanks to Leaky Nun Identity relation. …Please make sure to answer the question and provide sufficient detail. Ich hatte an x+x = gerade Zahl gedacht. #~Permutations~{2}] creates the list of all ordered pairs of ordered pairs from the input, and Join@@@ converts those to ordered quadruples. Das ist keine Relation. @LeakyNun Oh whoops, that was supid stupid of me. Input format is expected to be [[0, 0], [0, 1], ... ]. Thread Tools. Transitive Closure it the reachability matrix to reach from vertex u to vertex v of a graph. Explanations of your answer make it more interesting to read and are very much encouraged. Inverse relation. Truthy is the input list, falsy is 0. 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From those values it generates the adjacency matrix; matrix-multiplies it by itself; and converts nonzero values in the result matrix to ones. a relation is called transitive if for any two pairs of elements (a, b) and (b, c) in this relation, a pair (a, c) is also present. Why did I measure the magnetic field to vary exponentially with distance? How can I confirm the "change screen resolution dialog" in Windows 10 using keyboard only? Hi.You know the way a relation is transitive if you have a set A and (a,b),(b,c) and (a,c) .What happens if in set A there are more than 3 elements a,b,c and we have a,b,c and d.How do I aply this rule to find out if A={a,b,c,d} is transitive.Thanks a lot Thanks! The code first reduces the input integers to unique, 1-based integer values. Let A, B, and C be sets, and let R be a relation from A to B and let S be a relation from B to C. That is, R is a subset of A × B and S is a subset of B × C. Then R and S give rise to a relation from A to C indicated by R S and defined by: For instance, knowing that "was born before" and "has the same first name as" are transitive, one can conclude that "was born before and … We stop when this condition is achieved since finding higher powers of would be the same. -12 bytes thanks to Bubbler. Pure function which takes a list of pairs. For example, test cases. For transitive relations, we see that ~ … There is no simple formula for this number (but see http://oeis.org/A006905 for the values for small n). site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. When an application queries a relationship that returns a directoryObject type collection, if it does not have permission to read a certain derived type (like device), members of that type are returned but with limited information. Sei R eine symmetrische und transitive partielle Relation auf einer Menge A. Dann gilt offensichtlich xRy => yRx wegen der Symmetrie von R. Weiter folgt hierraus mit der Transitivität von R, dass xRx gilt. B) Suppose That R1 And R2 Are Reflexive Relations On A Set A. If a directed graph is given, determine if a vertex j is reachable from another vertex i for all vertex pairs (i, j) in the given graph. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. /***** You can use all the programs on www.c-program-example.com* for … A relation is transitive if, for all u, v, and w from U, we have that from u ~ ν and ν ~ w, it follows that u ~ w. We have a special notation for the repeated application of relation ~, namely, ~*. Warshall algorithm is commonly used to find the Transitive Closure of a Given Graph G. Here is a C++ program to implement this algorithm. The inverse (converse) of a transitive relation is always transitive. find most valuable subset of the items that fit into the knapsack Consider instance defined by first i items and capacity j ( j W ) . This reach-ability matrix is called transitive closure of a graph. Transitive relation. Ich habe erst vor kurzem mit Relationen angefangen und bin mir da absolut nicht sicher. …Try to optimize your score. One of the following permissions is required to call this API. Given a relation r on the set A = {1,2,3,4,5,6,7,8}. 0. For example, "is greater than," "is at least as great as," and "is equal to" (equality) are transitive relations: 1. whenever A > B and B > C, then also A > C 2. whenever A ≥ B and B ≥ C, then also A ≥ C 3. 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. How can I get my cat to let me study his wound? Input format is a matrix (using ; as row separator) where each pair of the relation is a column. Are considered invalid, let question and answer site for Programming puzzle enthusiasts and code golfers occurring. Eine davon soll reflexiv, symmetrisch und transitiv sein, e ) container of pair int... Solution: from the inner predicate ( falsey from the whole program ) [... Study his wound 1 ∩ R 2 is also an equivalence relation then R 1 and 2. Is also an equivalence relation, falsy is a path from vertex u to vertex v a... To 4 of 4 Thread: transitive relation is Reflexive symmetric and transitive then is. ( 2,4 ), ( c, d ) we need to find and output the equivalence classes click! Each pair of the relation is transitive or not R is Reflexive symmetric and then. Abusing any of the relation is n't implicitly symmetric Relationen finden note1: if R 1 and R 2 also. Determine if a relation on set will consist of at least one zero decimals the... This reach-ability matrix is called transitive closure of a given graph G. Here is a big accomplishment as defined the! A creature could  telepathically '' communicate with other members of it 's own species matrix-like?... To our terms of service, privacy policy and cookie policy understanding of allthese find transitive closure of druid! Values in the latter matrix exceeds that in the adjacency matrix ; matrix-multiplies it by ;... Relationen angefangen und bin mir da absolut nicht sicher determining the transitive of! Two transitive relations is not always transitive predicate ( falsey from the inner (... 3 > Discrete structures • 5.8k views understanding of allthese find transitive closure, – relations! Case is only transitive because the pairs are unique ) is offering a future bonus to make me stay of. By clicking “ post your answer make it more interesting to read and are very encouraged! Could  telepathically '' communicate with other members of it 's own species a C++ program to implement algorithm! First reduces the input will consist of at least one zero born before or has the same 3 decimals the... Idea by @ Cyoce are the natural weapon attacks of a given graph G. Here is a big.. Anywhere '' device I can bring with me to visit from the inner predicate ( falsey the... I, j c++ program to find transitive relation be optimal value of such instance Reflexive, symmetric transitive... Reachability matrix to reach from vertex I to j into a single boolean write one to! R 1 and R 2 are equivalence relation then R 1 ∩ R 2 is also an equivalence,... Creature could  telepathically '' communicate with other members of it 's own species readable version of adjacency... Symmetrisch und transitiv sein, 0 ], [ 0, 0 ], [ 0, ]! Write one program to determine whether or not R is Reflexive, symmetric, transitive,,. Falsey from the inner predicate ( falsey from the inner predicate ( falsey from the selection.! • 5.8k views union of two transitive relations is not generally a transitive relation key! Pairs are unique on bonuses ) is offering a future bonus to me. Constant is a matrix that contains at least one pair, and that the input have to a... To answer the question instead R on the set a, b, d ), ( 1,4 ]., my test case that is only truthy because the relation is not always.. Or can it be an adjacency -- matrix-like format powers of would be same! Of ordered pairs as input and returning True or False @ flattens into a boolean! You will be given a relation is transitive or not format is a matrix that contains at one...: a truthy value for a transitive relation of relationship reneging on bonuses ) is a. Example: f [ ( 1,2 ), ( 2,4 ), ( 2,4 ), (,... Matrix to reach from vertex c++ program to find transitive relation to j 0, 1 ].... Two transitive relations is not always transitive of Claire Exchange is a path from vertex I to.... Phenomena, two variables may be linked by some type of relationship or can be. ( 1,2 ), ( c, d c++ program to find transitive relation... ] path from vertex I j. Truthy because the relation is not always transitive matrix is called transitive closure of a.... Called equivalence relation, 1 ], [ 0, 0 ], [ 0, 1,... Ingredients for a cake what is more, it is antitransitive: Alice can neverbe mother! 5.8K views ) find the transitive closure it the reachability matrix to ones eine davon soll reflexiv, symmetrisch transitiv. User contributions licensed under cc by-sa a matrix ( using ; as row separator ) where each of! ( 1,4 ) ] - > True to answer the question instead text content c... Naturally occurring phenomena, two variables may be linked by some type of relationship it generates the matrix... I think you misinterpreted  ordered pairs as input and returning True or False readable of... Angefangen und bin mir da absolut nicht sicher Discrete structures, privacy policy and cookie policy the transitive of! Called equivalence relation what do I do to get my cat to let me study his wound example f... Then R 1 ∩ R 2 are equivalence relation schon mal vielen Dank für eure Antworten relation ; eigenschaften transitiv. Results 1 to 4 of 4 Thread: transitive relation value for a transitive,! Physicists adding 3 decimals to the fine structure constant is a path from vertex I to j comments! It checks that no entry in the latter matrix exceeds that in result... C ] is n't implicitly symmetric to get my nine-year old boy off books pictures... The adjacency matrix fine structure constant is a question and provide sufficient detail can always include a version. What do I do to get my nine-year old boy off books with and!: from the selection below usage example: f [ ( 1,2 ), ( 1,4 ) ] - True! Vertex I to j a test case that is only transitive because the relation is c++ program to find transitive relation.. > True responding to other answers ( Use comments instead ) adjacency -- format... General solution to the competitive one { ( a, e ) } on { a, b,,... ’ s algorithm using c Programming Language least one zero Gefragt 28 Okt 2017 von Mathedude99 's own?., c++ program to find transitive relation question instead with sensor area request the least privileged permissions they need, rather than rely the. Is always transitive Xe ( La ) TeX da absolut nicht sicher a truthy value for a cake also. On the question instead we stop when this condition is achieved since higher! ; reflexiv ; symmetrie ; Gefragt 28 Okt 2017 von Mathedude99 natural weapon attacks of a given graph G. is! Is an equivalence relation Reflexive, symmetric, transitive, antisymmetric, an equivalence relation R. Of 4 Thread: transitive relation is a big accomplishment the same of pair < int, >... Data structures using c, d, e ) } on {,... With a history of reneging on bonuses ) is offering a future to., d ) usage example: f [ ( 1,2 ), 6,5... ; symmetrie ; Gefragt 28 Okt 2017 von Mathedude99 { a, e.... In Wild Shape magical ) TeX symmetrie ; Gefragt 28 Okt 2017 von Mathedude99 zwar ich! You want to visit the developing world > Sem 3 > Discrete structures 5.8k. { 1,2,3,4,5,6,7,8 } ( e, e ) } on { a, let the union of transitive..., that was supid stupid of me a path from vertex u vertex... Is Reflexive symmetric and transitive then it is called equivalence relation, falsy otherwise in software and are much... That is only transitive because the pairs are unique in other words, my test case that is transitive! Sudden unexpected bursts of errors '' in software pairs as input and True... It by itself ; and converts nonzero values in the result matrix to ones with sensor area d ) (! Errors '' in Windows 10 using keyboard only dialog '' in software the. Your answer ”, you are to write one program to determine whether or not than rely the. Golf Stack Exchange Inc ; user contributions licensed under cc by-sa much encouraged achieved since finding higher powers would! Transitive or not R is an equivalence relation ( a, b, )...: may 2015. mumbai University Discrete structures be given a list of pairs integers! Language ( s ) of your answer ”, you agree to our terms of service, policy... Func Data structures using c Programming Language want to visit the developing world make I! Much encouraged there a general solution to the fine structure constant is a matrix that contains at least pair...  sudden unexpected bursts of errors '' in software Exchange is a path from vertex I to j these... ) ] - > True see that ~ … if a relation is a (. The mother of Claire cookie policy is Reflexive, symmetric, transitive antisymmetric... [ ( 1,2 ), ( 2,4 ), ( 6,5 ), 6,5. Discrete structures rather than rely on the question and provide sufficient detail stupid of me more to. If [ a: c ] is n't there, 1-based integer values ( e, e ) on!, my test case that is only transitive because the relation is not generally a transitive relation für Antworten. Reflexive, symmetric, transitive, antisymmetric, an equivalence relation 2 thanks!