Transitive and Nontransitive Relationships

Transitive and Nontransitive Relationships

The relationship R is called transitive if ARB and BRC produce ARC. For example, the relationship faster is transitive. Examples of transitive relationships: one can achieve, more, less, greater or equal, less or equal, depends on, the subset, is in the projection. If A is faster than B and B is faster than C, then, consequently, A is faster than C.


A relationship that is not transitive is nontransitive. Examples of nontransitive relationships: friend, neighbor, lies, is related (directly). For example, it does not follow from the fact that A lies to C  because A lies to B and B lies to C. Actually, A could lie to C, but this does not in any way follow from the fact that A lied to B and B lied to C. For example, we know that A did not speak to C.


Although in most cases it is easy to understand whether the relationships are transitive or not, this is not always the case in general. An example of a transitive game is the children’s game “Stone, Scissors, Paper“. In this game, scissors cut paper, paper covers a stone, and a stone dulls scissors. The matter is complicated by the fact that all three relations (cuts, covers, dulls) are equivalent in this case to a relationship stronger than. Many other children’s and adult games have also been developed based on the ambiguity of transitivity.


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It is in our interests (since this is one of the testing tasks) to verify that all relationships which must be transitive are really transitive, and vice versa, if transitivity is not a property of the relationship, then it is necessary to verify that it is not intransitive. This is not always obvious in natural languages, so the specifications can be misleading. For example, a common relationship associated in a natural language can be interpreted ambiguously. Does this mean that node B can be reached from node A, or does it mean that A is directly connected to B? In the first case, the relationship can be reached must be transitive. This means that if B can be reached from A, and C can be reached from B, then one can reach C from A. But the relationship can be reached is not necessarily transitive. What if we add “touching the hand”? For example, I can reach my neighbor by touching her hand, and she, in turn, can reach her neighbor by touching her arm, but this does not mean that I can reach her neighbor by touching her hand if I, of course, not an orangutan or if we do not sit very closely with her.


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