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Anti-symmetric and transitive properties in relations

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Steinhart's definition of anti-symmetry is as follows: "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). "Part"- relationships are anti-symmetric. If J is a part of K, and K is a part of J, then the two are identical. This is not always the case, however:

"Note that anti-symmetry and symmetry are not opposites. There are relations that are neither symmetric nor anti-symmetric. Consider the relation is-at-least-as-old-as. Since there are many distinct people with the same age, there are cases in which x and y are distinct; x is at least as old as y; and y is at least as old as x. There are cases in which (x, y) and (y, x) are in the relation but x is not identical to y. Thus the relation is not anti-symmetric. But for any x and y, the fact that x is at least as old as y does not imply that y is at leaset as old as x. Hence the relation is not symmetric"(Steinhart, p. 26).

Transitivity can be understood in terms of the an argument of the logical form "if A is larger than B, and B is larger than C, then A is larger than C" (Steinhart, p. 27). Steinhart articulates the definition of transitivity thus: "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. 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, p. 27).

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

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