Transitive relation

Definition

A homogeneous relation R on the set X is a transitive relation if, :for all a, b, c ∈ X, if a R b and b R c, then a R c. Or in terms of first-order logic: :\forall a,b,c \in X: (aRb \wedge bRc) \Rightarrow aRc, where a R b is the infix notation for (a, b) ∈ R.

Examples

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:

: whenever x y and y z, then also x z : whenever x ≥ y and y ≥ z, then also x ≥ z : whenever x = y and y = z, then also x = z.

More examples of transitive relations:

• "is a subset of" (set inclusion, a relation on sets)
• "divides" (divisibility, a relation on natural numbers)
• "implies" (implication, symbolized by "⇒", a relation on propositions)

Examples of non-transitive relations:

• "is the successor of" (a relation on natural numbers)
• "is a member of the set" (symbolized as "∈")
• "is perpendicular to" (a relation on lines in Euclidean geometry)

The empty relation on any set X is transitive because there are no elements a,b,c \in X such that aRb and bRc, 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 (x, x) for some x \in X the only such elements a,b,c \in X are a=b=c=x, and indeed in this case aRc, while if the ordered pair is not of the form (x, x) then there are no such elements a,b,c \in X and hence R is vacuously transitive.

Vacuous transitivity is transitivity when in a relation there are no ordered pairs of the form (a,b) and (b,c).

Properties

Closure properties

• The converse (inverse) of a transitive relation is always transitive. For instance, knowing that "is a subset of" is transitive and "is a superset of" is its converse, one can conclude that the latter is transitive as well.
• The intersection of two transitive relations is always transitive. For instance, knowing that "was born before" and "has the same first name as" are transitive, one can conclude that "was born before and also has the same first name as" is also transitive.
• The union of two transitive relations need not be transitive. For instance, "was born before or has the same first name as" is not a transitive relation, since e.g. Herbert Hoover is related to Franklin D. Roosevelt, who is in turn related to Franklin Pierce, while Hoover is not related to Franklin Pierce.
• The complement of a transitive relation need not be transitive. For instance, while "equal to" is transitive, "not equal to" is only transitive on sets with at most one element.

Other properties

A transitive relation is asymmetric if and only if it is irreflexive.

A transitive relation need not be reflexive. When it is, it is called a preorder. For example, on set X = {1,2,3}:

• R = {(1,1), (2,2), (3,3), (1,3), (3,2)} is reflexive, but not transitive, as the pair (1,2) is absent,
• R = {(1,1), (2,2), (3,3), (1,3)} is reflexive as well as transitive, so it is a preorder,
• R = {(1,1), (2,2), (3,3)} is reflexive as well as transitive, another preorder,
• R = {(1,2), (2,3), (1,3)} is transitive, but not reflexive.

As a counter example, the relation on the real numbers is transitive, but not reflexive.

Transitive extensions and transitive closure

Main article: Transitive closure

Let R be a binary relation on set X. The transitive extension of R, denoted R1, is the smallest binary relation on X such that R1 contains R, and if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R1. 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) ∈ R1 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 .

The transitive extension of R1 would be denoted by R2, 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, R1, R2, ... .

The transitive closure of a relation is a transitive relation.

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.

Relation types that require transitivity

• Preorder – a reflexive and transitive relation
• Partial order – an antisymmetric preorder
• Total preorder – a connected (formerly called total) preorder
• Equivalence relation – a symmetric preorder
• Strict weak ordering – a strict partial order in which incomparability is an equivalence relation
• Total ordering – a connected (total), antisymmetric, and transitive relation

Counting transitive relations

No general formula that counts the number of transitive relations on a finite set is known. However, there is a formula for finding the number of relations that are simultaneously reflexive, symmetric, and transitive – in other words, equivalence relations – , those that are symmetric and transitive, those that are symmetric, transitive, and antisymmetric, and those that are total, transitive, and antisymmetric. Pfeiffer{{citation | article-number = 04.3.2 | pages = 1–11 | article-number = 05.2.1 | pages = 1–7

Since the reflexivization of any transitive relation is a preorder, the number of transitive relations an on n-element set is at most 2n time more than the number of preorders, thus it is asymptotically 2^{(1/4+o(1))n^2} by results of Kleitman and Rothschild.

Related properties

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, but not antitransitive. The relation defined by xRy if x is even and y is odd is both transitive and antitransitive. The relation defined by xRy if x is the successor number of y is both intransitive and antitransitive. Unexpected examples of intransitivity arise in situations such as political questions or group preferences.

Generalized to stochastic versions (stochastic transitivity), the study of transitivity finds applications of in decision theory, psychometrics and utility models.

A quasitransitive relation is another generalization; it is required to be transitive only on its non-symmetric part. Such relations are used in social choice theory or microeconomics.

Proposition: If R is a univalent, then R;RT is transitive. : proof: Suppose x R;R^T y R;R^T z. Then there are a and b such that x R a R^T y R b R^T z . Since R is univalent, yRb and aRTy imply a=b. Therefore xRaRTz, hence xR;RTz and R;RT is transitive.

Corollary: If R is univalent, then R;RT is an equivalence relation on the domain of R. : proof: R;RT is symmetric and reflexive on its domain. With univalence of R, the transitive requirement for equivalence is fulfilled.

Notes

References

References

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2. However, the class of [[von Neumann ordinal]]s is constructed in a way such that ∈ ''is'' transitive when restricted to that class.
3. {{harvnb. Smith. Eggen. St. Andre. 2006
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11. since e.g. 3''R''4 and 4''R''5, but not 3''R''5
12. since e.g. 2''R''3 and 3''R''4 and 2''R''4
13. since ''xRy'' and ''yRz'' can never happen
14. since e.g. 3''R''2 and 2''R''1, but not 3''R''1
15. since, more generally, ''xRy'' and ''yRz'' implies ''x''=''y''+1=''z''+2≠''z''+1, i.e. not ''xRz'', for all ''x'', ''y'', ''z''
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