Generic Property

Generic Property

In mathematics, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functions, as in the statements, "A generic polynomial does not have a root at zero," or "A generic matrix is invertible." As another example, a generic property of a space is a property that holds at "almost all" points of the space, as in the statement, "If f : M → N is a smooth function between smooth manifolds, then a generic point of N is not a critical value of f." (This is by Sard's theorem.)

There are many different notions of "generic" (what is meant by "almost all") in mathematics, with corresponding dual notions of "almost none" (negligible set); the two main classes are:

  • In measure theory, a generic property is one that holds almost everywhere, meaning "with probability 1", with the dual concept being null set, meaning "with probability 0".
  • In topology and algebraic geometry, a generic property is one that holds on a dense open set, or more generally on a residual set, with the dual concept being a nowhere dense set, or more generally a meagre set.

Read more about Generic Property:  Definitions: Measure Theory, Definitions: Topology, Genericity Results

Famous quotes containing the words generic and/or property:

    ““Mother” has always been a generic term synonymous with love, devotion, and sacrifice. There’s always been something mystical and reverent about them. They’re the Walter Cronkites of the human race . . . infallible, virtuous, without flaws and conceived without original sin, with no room for ambivalence.”
    —Erma Bombeck (20th century)

    “To throw obstacles in the way of a complete education is like putting out the eyes; to deny the rights of property is like cutting off the hands. To refuse political equality is like robbing the ostracized of all self-respect, of credit in the market place, of recompense in the world of work, of a voice in choosing those who make and administer the law, a choice in the jury before whom they are tried, and in the judge who decides their punishment.”
    —Elizabeth Cady Stanton (1815–1902)