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 merely defines the extension of property and says nothing about what causes it to hold for exactly those values.
