1. Overview
This summary table gathers the key properties of relations in one place. It shows what each property means and how it behaves. This helps compare different properties at a glance.
2. Summary Table of Relation Properties
| Property | Condition | Main Idea |
|---|---|---|
| Reflexive | (a,a) ∈ R for all a | Every element relates to itself. |
| Irreflexive | (a,a) ∉ R for all a | No element relates to itself. |
| Symmetric | (a,b) ∈ R ⇒ (b,a) ∈ R | Relations work both ways. |
| Antisymmetric | (a,b),(b,a) ∈ R ⇒ a = b | Reverse pairs allowed only if elements are equal. |
| Asymmetric | (a,b) ∈ R ⇒ (b,a) ∉ R (a,a) ∉ R | Strictly one-way; no reverse pairs and no self-pairs. |
| Transitive | (a,b),(b,c) ∈ R ⇒ (a,c) ∈ R | Links pass through a middle element. |
3. How to Use This Table
Each property adds a different structure to a relation. By comparing these conditions, it becomes easier to classify a relation and understand how it behaves. This table is helpful when studying equivalence relations or partial orders.