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.
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:
∀ a , b , c ∈ X : ( a R b ∧ b R c ) ⇒ a R c {\displaystyle \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 {\displaystyle X} is transitive because there are no elements a , b , c ∈ X {\displaystyle a,b,c\in X} such that a R b {\displaystyle aRb} and b R c {\displaystyle 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 ) {\displaystyle (x,x)} for some x ∈ X {\displaystyle x\in X} the only such elements a , b , c ∈ X {\displaystyle a,b,c\in X} are a = b = c = x {\displaystyle a=b=c=x} , and indeed in this case a R c {\displaystyle aRc} , while if the ordered pair is not of the form ( x , x ) {\displaystyle (x,x)} then there are no such elements a , b , c ∈ X {\displaystyle a,b,c\in X} and hence R {\displaystyle 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 < {\displaystyle <} on the real numbers is transitive, but not reflexive.
Transitive extensions and transitive closure
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