Can somebody tell me if this is correct? Let's say we are working with the set A = {a, b, c, d} and the relation R comes from A.
Reflexive: the relation is reflexive if and only if all of (a, a), (b, b), (c, c) and (d, d) are present. If at least one of them is missing, it is not reflexive.
Symmetric: the relation is symmetric if and only if all ordered pairs present within the relation have a symmetrically matching ordered pair present as well. For example, {(a, b), (b, a)} is symmetric (even though c and d are never used) but {(a, b), (b, a), (a, c)} is not symmetric.
Antisymmetric: the relation is symmetric if and only if no symmetry exists. The exception to this rule is in the case of ordered pairs such as (a, a) and (b, b), etc. For example, {(a, b), (a, c)} is antisymmetric because no symmetry exists, but {(a, a), (b, b), (c, c)} is both symmetric and antisymmetric.
Transitive: the relation is transitive if and only if all possibilities for transitivity exist for every ordered pair that is present in the relation. For example, {(a, b), (b, c), (a, c)} is transitive, but {(a, b), (b, c), (a, c), (c, b), (b, d)} is not because (c, b) and (b, d) exist, but not (c, d).
Do I understand these rules correctly? Thank you for any clarification.
I’d express a few things a bit differently, but the substance is fine. (For instance, I wouldn’t say that an ordered pair exists, but rather that it’s in the relation.)