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Concept
Intransitivity
Summary
In mathematics, intransitivity (sometimes called nontransitivity) is a property of binary relations that are not transitive relations. This may include any relation that is not transitive, or the stronger property of antitransitivity, which describes a relation that is never transitive.
A relation is transitive if, whenever it relates some A to some B, and that B to some C, it also relates that A to that C. Some authors call a relation if it is not transitive, that is, (if the relation in question is named )
This statement is equivalent to
For example, consider the relation R on the integers such that a R b if and only if a is a multiple of b or a divisor of b. This relation is intransitive since, for example, 2 R 6 (2 is a divisor of 6) and 6 R 3 (6 is a multiple of 3), but 2 is neither a multiple nor a divisor of 3. This does not imply that the relation is (see below); for example, 2 R 6, 6 R 12, and 2 R 12 as well.
As another example, in the food chain, wolves feed on deer, and deer feed on grass, but wolves do not feed on grass. Thus, the relation among life forms is intransitive, in this sense.
Another example that does not involve preference loops arises in freemasonry: in some instances lodge A recognizes lodge B, and lodge B recognizes lodge C, but lodge A does not recognize lodge C. Thus the recognition relation among Masonic lodges is intransitive.
Often the term is used to refer to the stronger property of antitransitivity.
In the example above, the relation is not transitive, but it still contains some transitivity: for instance, humans feed on rabbits, rabbits feed on carrots, and humans also feed on carrots.
A relation is if this never occurs at all, i.e.
Many authors use the term to mean .
For example, the relation R on the integers, such that a R b if and only if a + b is odd, is intransitive. If a R b and b R c, then either a and c are both odd and b is even, or vice-versa. In either case, a + c is even.
A second example of an antitransitive relation: the defeated relation in knockout tournaments.
Official source
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