Reflexive, Symmetric and Transitive Relations
A relation on is reflexive when , symmetric when , and transitive when , in each case for all .
Definition 2.28 (Relation Reflexitivity, Symmetry and Transitivity)
A relation on is…
- reflexive if , .
- symmetric if , .
- transitive if , .
Checking the examples
For the five relations on sharing a final digit, , , equality to on both sides, and : the final-digit relation and are reflexive and symmetric, though only the former is transitive; is transitive only; is symmetric only; and exactly when is transitive only. Only the final-digit relation satisfies all three properties.