1. What a Universal Relation Means
A universal relation contains every possible ordered pair from the Cartesian product. This means every element is related to every element. Nothing is left out.
It represents the idea of “everything is connected to everything”.
2. Formal Definition
If a relation is defined on a set A, then the universal relation on A is:
R = A \times A
Every possible pair appears in the relation.
3. Why It Is Called “Universal”
The relation includes all possible combinations, so it covers the entire Cartesian product. There are no restrictions — every element connects to every element.
3.1. Illustration
A = \{1,2\}
A \times A = \{(1,1),(1,2),(2,1),(2,2)\}
Universal relation: R = \{(1,1),(1,2),(2,1),(2,2)\}
4. Properties of the Universal Relation
Since all pairs are included, several properties automatically hold.
4.1. Observations
- Reflexive (all self-pairs like (a,a) are present).
- Symmetric (if (a,b) is present, (b,a) is also present).
- Transitive (every possible pair is already included).
- Not antisymmetric unless the set has only one element.
- Not irreflexive because self-pairs always exist.
5. Example for Understanding
If A = \{a,b,c\}, then the universal relation is:
R = \{(a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,a),(c,b),(c,c)\}
Every element is linked with every element.