Transitive Property

Author: Ernad Mujakic
Date: 2025-07-25

If a relation is transitive then for all possible elements , and , if that relation holds for and , and that same relation holds between and , then that relation must hold between and .

Where:

  • is some relation such as equality or inequality.

Relations

  • Equality: If and then .
  • Inequality: If and then .
  • Set Inclusion: If and then .
  • Implication: If and then .
  • Inheritance: If inherits from and inherits from then inherits from .

Applications

  • Mathematics: Used extensively in algebra, geometry, and set theory to establish relationships between numbers, variables, or sets.
  • Computer Science: Used in graph theory algorithms like Warshall's Algorithm which is used to determine the Transitive Closure of a directed graph. The Transitive Closure of a directed graph represents which vertices are reachable from others. The Transitive Reduction of a directed graph is the smallest reduction that has the same reachability relation as the original graph.
  • Logic: The transitive property is a fundamental property in proofs and reasoning, allowing for the derivation of conclusions based on established relationships.

Reflexive Property

The reflexive property states that any value or expression is equal to itself. The equality relation is an example of a reflexive operation, since all real numbers or variables are equal to themselves.

Symmetric Property

A relation is symmetric if for all possible elements and , if is related to , then is related to . Equality is an example of a symmetric relation.

Equivalence Relation

An equivalence relation is a relation that is reflexive, symmetric, and transitive. Equivalence relations can group elements of a set into distinct categories called Equivalence Classes, which are disjoint subsets containing elements that are equivalent to one another.

Preorder & Partial Order

A partial order on a set is a binary relation that is reflexive, antisymmetric, and transitive. It is a way to order elements in a set where not all pairs of elements need to be comparable. Partial orders are visualized using a Hasse Diagram.

Preorder is a binary relation defined on a set that is both reflexive and transitive. It is a generalization of partial orders in the sense that it does not require antisymmetry. Preorders are commonly applied in Decision Theory to model preferences, or in Game Theory to compare strategies.

Intransitivity & Antitransitivity

  • A binary relation is intransitive if there exists three values where transitivity does not hold.
  • Antitransitivity is a stronger property which holds if for any three values, transitivity never holds.

References