Transitive relation

A relation is transitive when, for all elements a, b, and c, if it relates a to b and b to c, it also relates a to c.

Binary relation: A set of ordered pairs that specifies which elements are related. Transitivity is a property of binary relations between elements.
Transitive closure: The smallest transitive relation containing a given relation. It adds precisely the links needed to make a relation transitive.
Reflexive relation: A relation in which every element is related to itself. Transitivity does not require self-relations, so reflexivity is an independent property.
Nontransitive relation: A relation for which some related pair of steps fail to imply the corresponding endpoint relation. A missing endpoint link is exactly what violates transitivity.
Ordered pair: A pair of objects in which the first and second positions are distinguished. A relation records its links as ordered pairs, such as (a,b) and (b,c).
Directed graph: A graph whose edges have specified directions between vertices. A relation can be drawn as directed edges, making two-step paths and required shortcut edges visible.
Symmetric relation: A relation in which reversing any related pair preserves the relation. Symmetry and transitivity are separate conditions, even when both hold together.
Intransitive relation: A relation in which no three elements satisfy both successive links and the endpoint link. Unlike mere failure of transitivity, intransitivity rules out every such three-link pattern.
Cartesian product: The set of all ordered pairs formed from two specified sets. A relation on a set is a subset of that set's Cartesian product.
Reachability: The property that a path connects one vertex to another in a graph. Reachability is transitive because paths can be concatenated.

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