We have been considering special properties that a dyadic relation may have, in particular, the following two symmetry properties.
- A dyadic relation is symmetric if being in implies that is in
- A dyadic relation is asymmetric if being in implies that is not in
The first thing to understand about any symmetry of any relation is that it is a property of the whole relation, the whole set of tuples, not a property of individual tuples.
Many properties of dyadic relations can be made visually evident by arranging their ordered pairs in 2-dimensional arrays. Let’s do this for our initial sample of biblical brothers, using the first three letters of their names as row and column labels.
The relation indicated by “brother of” is a symmetric relation. The ordered pairs of are given below.
The relation indicated by “elder brother of” is an asymmetric relation. The ordered pairs of are given below.