Type | Binary relation |
---|---|
Field | Elementary algebra |
Statement | A relation on a set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . |
Symbolic statement |
In mathematics, a binary relation R on a set X is transitive if, for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c.
Every partial order and every equivalence relation is transitive. For example, less than and equality among real numbers are both transitive: If a < b and b < c then a < c; and if x = y and y = z then x = z.
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indicates that the column's property is always true the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation be transitive: for all if and then |
A homogeneous relation R on the set X is a transitive relation if,^{[1]}
Or in terms of first-order logic:
where a R b is the infix notation for (a, b) ∈ R.
As a non-mathematical example, the relation "is an ancestor of" is transitive. For example, if Amy is an ancestor of Becky, and Becky is an ancestor of Carrie, then Amy is also an ancestor of Carrie.
On the other hand, "is the birth mother of" is not a transitive relation, because if Alice is the birth mother of Brenda, and Brenda is the birth mother of Claire, then it does not follow that Alice is the birth mother of Claire. In fact, this relation is antitransitive: Alice can never be the birth mother of Claire.
Non-transitive, non-antitransitive relations include sports fixtures (playoff schedules), 'knows' and 'talks to'.
The examples "is greater than", "is at least as great as", and "is equal to" (equality) are transitive relations on various sets. As are the set of real numbers or the set of natural numbers:
More examples of transitive relations:
Examples of non-transitive relations:
The empty relation on any set is transitive^{[3]} because there are no elements such that and , and hence the transitivity condition is vacuously true. A relation R containing only one ordered pair is also transitive: if the ordered pair is of the form for some the only such elements are , and indeed in this case , while if the ordered pair is not of the form then there are no such elements and hence is vacuously transitive.
A transitive relation is asymmetric if and only if it is irreflexive.^{[6]}
A transitive relation need not be reflexive. When it is, it is called a preorder. For example, on set X = {1,2,3}:
Let R be a binary relation on set X. The transitive extension of R, denoted R_{1}, is the smallest binary relation on X such that R_{1} contains R, and if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R_{1}.^{[7]} For example, suppose X is a set of towns, some of which are connected by roads. Let R be the relation on towns where (A, B) ∈ R if there is a road directly linking town A and town B. This relation need not be transitive. The transitive extension of this relation can be defined by (A, C) ∈ R_{1} if you can travel between towns A and C by using at most two roads.
If a relation is transitive then its transitive extension is itself, that is, if R is a transitive relation then R_{1} = R.
The transitive extension of R_{1} would be denoted by R_{2}, and continuing in this way, in general, the transitive extension of R_{i} would be R_{i + 1}. The transitive closure of R, denoted by R* or R^{∞} is the set union of R, R_{1}, R_{2}, ... .^{[8]}
The transitive closure of a relation is a transitive relation.^{[8]}
The relation "is the birth parent of" on a set of people is not a transitive relation. However, in biology the need often arises to consider birth parenthood over an arbitrary number of generations: the relation "is a birth ancestor of" is a transitive relation and it is the transitive closure of the relation "is the birth parent of".
For the example of towns and roads above, (A, C) ∈ R* provided you can travel between towns A and C using any number of roads.
No general formula that counts the number of transitive relations on a finite set (sequence A006905 in the OEIS) is known.^{[9]} 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 the OEIS), those that are symmetric and transitive, those that are symmetric, transitive, and antisymmetric, and those that are total, transitive, and antisymmetric. Pfeiffer^{[10]} 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 Brinkmann and McKay (2005).^{[11]}
Since the reflexivization of any transitive relation is a preorder, the number of transitive relations an on n-element set is at most 2^{n} time more than the number of preorders, thus it is asymptotically by results of Kleitman and Rothschild.^{[12]}
Elements | Any | Transitive | Reflexive | Symmetric | Preorder | Partial order | Total preorder | Total order | Equivalence relation |
---|---|---|---|---|---|---|---|---|---|
0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 |
2 | 16 | 13 | 4 | 8 | 4 | 3 | 3 | 2 | 2 |
3 | 512 | 171 | 64 | 64 | 29 | 19 | 13 | 6 | 5 |
4 | 65,536 | 3,994 | 4,096 | 1,024 | 355 | 219 | 75 | 24 | 15 |
n | 2^{n2} | 2^{n(n−1)} | 2^{n(n+1)/2} | ∑^{n} _{k=0} k!S(n, k) |
n! | ∑^{n} _{k=0} S(n, k) | |||
OEIS | A002416 | A006905 | A053763 | A006125 | A000798 | A001035 | A000670 | A000142 | A000110 |
Note that S(n, k) refers to Stirling numbers of the second kind.
A relation R is called intransitive if it is not transitive, that is, if xRy and yRz, but not xRz, for some x, y, z. In contrast, a relation R is called antitransitive if xRy and yRz always implies that xRz does not hold. For example, the relation defined by xRy if xy is an even number is intransitive,^{[13]} but not antitransitive.^{[14]} The relation defined by xRy if x is even and y is odd is both transitive and antitransitive.^{[15]} The relation defined by xRy if x is the successor number of y is both intransitive^{[16]} and antitransitive.^{[17]} Unexpected examples of intransitivity arise in situations such as political questions or group preferences.^{[18]}
Generalized to stochastic versions (stochastic transitivity), the study of transitivity finds applications of in decision theory, psychometrics and utility models.^{[19]}
A quasitransitive relation is another generalization;^{[5]} it is required to be transitive only on its non-symmetric part. Such relations are used in social choice theory or microeconomics.^{[20]}
Proposition: If R is a univalent, then R;R^{T} is transitive.
Corollary: If R is univalent, then R;R^{T} is an equivalence relation on the domain of R.