Appendices A.1 REVIEW OF GENERAL MATHEMATICAL NOTIONS ·························································································· · · · · · · · A relation is any subset of a Cartesian product. Definition 194: Relation ·························································································· · · · · · · · A binary relation from the set X to the set Y is a collection of ordered pairs (x, y) X × Y. · · One often writes xRy to show x is related to y. · Definition 195: Binary Relation ·························································································· · · · · · · · An equivalence relation on a set X is a subset of X 2 ; that is, a collection R of ordered pairs · · of elements of X which satisfies the following three properties: · · · · · 1. Reflexiveness: x X, xRx. · · 2. Symmetry: (x, y) X 2 , xRy yRx. · · · 3. Transitivity: (x, y, z) X 3 , xRy and yRz implies that xRz. · Definition 196: Equivalence Relation