Reflexive
A reflexive relation is one where every element is related to itself. For instance, in a reflexive relation R, every element has an ordered pair with itself. Therefore, if there exists even one ordered pair without the same elements, such as (b, b), the relation is not reflexive.
- Valid: A relation where every element is related to itself, such as R = {(1,1), (2,2), (3,3)}.
- Invalid: A relation where at least one element is not related to itself, such as R = {(1,1), (2,3)}.
Transitive
A transitive relation ensures that if there are ordered pairs (a, b) and (b, c) within the relation, then there must also be an ordered pair (a, c). If we find any pair (a, b) and (b, c) but not (a, c), the relation is not transitive.
- Valid: A relation where every time there are ordered pairs (a, b) and (b, c), there is also an ordered pair (a, c) within the relation, such as R = {(1,2), (2,3), (1,3)}.
- Invalid: A relation where there are ordered pairs (a, b) and (b, c) but no corresponding (a, c), such as R = {(1,2), (2,3)}.
Symmetric
A symmetric relation implies that for every ordered pair (a, b) in the relation, there exists an ordered pair (b, a). If there’s even one ordered pair (a, b) without a corresponding (b, a), the relation is not symmetric.
- Valid: A relation where for every ordered pair (a, b) in the relation, there exists an ordered pair (b, a), such as R = {(1,2), (2,1)}.
- Invalid: A relation where at least one ordered pair (a, b) doesn’t have a corresponding (b, a), such as R = {(1,2)}.
Anti-Symmetric
An anti-symmetric relation dictates that if there are ordered pairs (a, b) and (b, a) in the relation, where a ≠ b, then either (a, b) or (b, a) must be present, but not both. In simpler terms, if both pairs are present, a must equal b. If there are no such pairs (a, b) and (b, a), the relation satisfies anti-symmetry.
- Valid: A relation where for every ordered pair (a, b) and (b, a) present in the relation where a ≠ b, then either (a, b) or (b, a) must be in the relation, but not both, such as R = {(1,2), (2,3)}.
- Invalid: A relation where there are ordered pairs (a, b) and (b, a) present where a ≠ b, but both pairs are in the relation, such as R = {(1,2), (2,1)}.