In this article, we will give a brief, informal summary of the properties of relations in set theory which we surveyed before. This informal summary will consist simply of the examples given by Steinhart to illustrate each of the properties.
1) Reflexivity - "is-the-same-person-as." Something is the same thing as itself.
2) Symmetry - "is-married-to." For any x and y, if x bears some relation to y, then y bears that same relation to x.
3) Anti-Symmetry - "If Alpha is a part of Beta and Beta is a part of Alpha, then Alpha is identical with Beta"(Steinhart, p. 26). Stated formally, "A relation R on S is anti-symmetric iff for every x and y in S, if (x, y) is in R and (y, x) is in R, then x is identical to y"(Steinhart, p. 26).
4) Transitivity - "The relation is-taller-than is transitive. If Abe is taller than Ben, and Ben is taller than Carl, then Abe is taller than Carl"(Steinhart, pp. 26-27). Stated formally, "A relation R on S is transitive iff for every x, y, and z in S, if (x, y) is in R and (y, z) is in R, then (x, z) is in R"(Steinhart, p. 26).
Steinhart, Eric. "More Precisely: The Mathematics You Need To Do Philosophy." Broadview Press, 2009.