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Inverse of relations in set theory

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The "inverse" of a relation is simply when we turn it around. This relation is also known as a "converse" relation. For example, the inverse of the relation "is-faster-than" is "is-slower-than." For our purposes, the inverse of an ordered pair (x, y) is (y, x). The inverse relation is represented as an R with a subscript of "-1." We can represent this when we are without the luxury of superscripts as "R^-1."

Suppose we want to represent the inverse of (x, y) with set-builder notation. We would write something like this:

R^-1 = { (y, x) | (x, y) ∈ R}.

A description of the inverse of a relation in this case is as follows: there is an ordered pair (y, x) and its inverse is such that the original relation is the ordered pair (x, y).

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



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