in mathematics, a property is any characteristic that applies to a given set. rigorously, a property "p" defined for all elements of a set "x" is usually defined as a function "p": "x" → {true, false}, that is true whenever the property holds; or equivalently, as the subset of "x" for which "p" holds; i.e. the set {"x" | "p"("x") = true}; "p" is its indicator function. however, it may be objected that the rigorous definition defines merely the extension of a property, and says nothing about what causes the property to hold for exactly those values.