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Orders and quasi-orders in set theory

We speak of a relation as an "order relation" if and only if it is reflexive, symmetric and transitive (Steinhart, p. 44). One of the more well-known examples of an order relation is the relation "is-greater-than-or-equal-to" applied to a set of numbers (Steinhart, p. 44).

"Since R is reflexive, for all x in X, (x, x) is in R. Since R is anti-symmetric, for all x and y in X, if both (x, y) and (y, x) are in R, then x is identical with y. Since R is transitive, for all x, y, and z in X, if (x, y) is in R and (y, z) is in R, then (x, z) is in R"(Steinhart, p. 44).

There are also quasi-orders. These are reflexive and transitive, but they are not anti-symmetric. Steinhart's example is that of two people with the same birth dates. It is reflexive because a person is as old as he himself is. This is obvious. It is also transitive, because "If x is at least as old as y, and y is at least as old as z, then x is at least as old as z"(Steinhart, p. 44). Nevertheless, as Steinhart notes, this quasi-order is not anti-symmetric because one element being as old as the other does not necessarily entail that the two elements are identical, since they may be distinct people with the same birth dates (Steinhart, p. 44).

Steinhart, Eric. "More Precisely: The Mathematics You Need To Do Philosophy." Broadview Press, 2009.

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