University of California, San Diego: Edward Bender and S. Williamson's "Sets, Equivalence and Order: Sets and Functions"
Given a set A, a binary relation, R on A, is a subset of A x A. Note that a binary relation is a set of order pairs (x, y) where x and y are in A. The next subunits define properties of a binary relation, reflexive and symmetric. A third property is transitivity.
Relations are another critically important concept in set theory, functions, science, and every other subject. Work a lot of examples.
A binary relation on A is reflexive, if (a, a) ∈ R.
A binary relation is symmetric if (a, b) ∈ R, then (b, a) ∈ R.
A binary relation is transitive if (a, b) ∈ R, (b, c) ∈ R, then (a, c ) ∈ R. There is an important consequence of a binary relation being reflexive, symmetric, and transitive. A binary relation that has all 3 properties is called an equivalence relation. If a binary relation on A is an equivalence relation, it determines a partition of A. A partition of A is a set of subsets of A, which are mutually disjoint, and whose union is A.