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Reflexive and symmetric relations in set theory

Let's look at properties of relations in set theory. We will look at two for now:

1) Reflexivity

2) Symmetry

First, reflexivity. We speak of a relation R on a set S as "reflexive, if and only if, "for every x in S, (x, x) is in R"(Steinhart, p. 26). Reflexivity is exhibited in identity. Daniel is the same person as Daniel, for example, is a reflexive statement.

Next, symmetry. "A relation R on S is symmetric iff for every x and y in S, (x, y) is in R iff (y, x) is in R"(Steinhart, p. 26). To put it informally, a marriage relation would be one that in set theory, would be represented as a symmetric relationship. "For any x and y, if x is married to y, then y is married to x; and if y is married to x, then x is married to y. A symmetric relation is its own inverse. If R is symmetric, then R = R^-1"(Steinhart, p. 26).

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

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