# Category:Transitive Relations

This category contains results about transitivity of relations.

Let $\mathcal R \subseteq S \times S$ be a relation in $S$.

$\mathcal R$ is **transitive** if and only if:

- $\tuple {x, y} \in \mathcal R \land \tuple {y, z} \in \mathcal R \implies \tuple {x, z} \in \mathcal R$

that is:

- $\set {\tuple {x, y}, \tuple {y, z} } \subseteq \mathcal R \implies \tuple {x, z} \in \mathcal R$

## Subcategories

This category has the following 7 subcategories, out of 7 total.

### E

### I

### R

### T

## Pages in category "Transitive Relations"

The following 39 pages are in this category, out of 39 total.

### A

### I

### R

- Reflexive and Symmetric Relation is not necessarily Transitive
- Reflexive and Transitive Relation is Idempotent
- Reflexive and Transitive Relation is not necessarily Symmetric
- Reflexive Closure of Transitive Antisymmetric Relation is Ordering
- Reflexive Relation on Set of Cardinality 2 is Transitive
- Relation Induced by Strict Positivity Property is Transitive
- Relation is Reflexive Symmetric and Antisymmetric iff Diagonal Relation
- Relation Isomorphism Preserves Transitivity
- Relation on Set of Cardinality 1 is Symmetric and Transitive
- Relation on Set of Cardinality 2 cannot be Non-Symmetric and Non-Transitive
- Relations with Combinations of Reflexivity, Symmetry and Transitivity Properties
- Restriction of Antitransitive Relation is Antitransitive
- Restriction of Non-Transitive Relation is Not Necessarily Non-Transitive
- Restriction of Transitive Relation is Transitive