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Symmetric closure

It is possible to transform a non-symmetric relation into a symmetric one. We do this by "adding (x, y) to R iff (y, x) is already in R"(Steinhart, p. 29). This produces a new relation known as a symmetric closure. The symmetric closure of R is formally articulated thus:

R ∪ {(y, x) | (x, y) ∈ R}

It is obvious that this is a symmetric relation, keeping in mind the definition of symmetry, according to which "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).

{ (y, x) | (x, y) ∈ R} is the inverse of R. We signify this with R^-1. The symmetric closure of R is therefore the union of R and R^-1 (Steinhart, p. 30).

Steinhart summarizes:

"...suppose we have the set of people {Allan, Betty, Carl, Diane}. Within this set, Allan is the husband of Betty, and Carl is the husband of Diane. We thus have the non-symmetric relation

is-the-husband-of = {(Allan, Betty), (Carl, Diane)}.

We make this into the new symmetric relation is-married-to by taking the pairs in is-the-husband-of and adding pairrs of the form (wife, y, husband x) for each pair of the form (husband, x, wife y) in is-the-husband-of. We thus get the symmetric closure

is-married-to = {(Allan, Betty), (Carl, Diane), (Betty, Allan), (Diane, Carl)} (Steinhart, p. 30).

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