Naive Set Theory - Universal Sets and Absolute Complements

Universal Sets and Absolute Complements

In certain contexts we may consider all sets under consideration as being subsets of some given universal set. For instance, if we are investigating properties of the real numbers R (and subsets of R), then we may take R as our universal set. A true universal set is not included in standard set theory (see Paradoxes below), but is included in some non-standard set theories.

Given a universal set U and a subset A of U, we may define the complement of A (in U) as

AC := {xU : xA}.

In other words, AC ("A-complement"; sometimes simply A', "A-prime" ) is the set of all members of U which are not members of A. Thus with R, Z and O defined as in the section on subsets, if Z is the universal set, then OC is the set of even integers, while if R is the universal set, then OC is the set of all real numbers that are either even integers or not integers at all.

Read more about this topic:  Naive Set Theory

Famous quotes containing the words universal, sets and/or absolute:

    We call contrary to nature what happens contrary to custom; nothing is anything but according to nature, whatever it may be, Let this universal and natural reason drive out of us the error and astonishment that novelty brings us.
    Michel de Montaigne (1533–1592)

    I think middle-age is the best time, if we can escape the fatty degeneration of the conscience which often sets in at about fifty.
    —W.R. (William Ralph)

    Complete courage and absolute cowardice are extremes that very few men fall into. The vast middle space contains all the intermediate kinds and degrees of courage; and these differ as much from one another as men’s faces or their humors do.
    François, Duc De La Rochefoucauld (1613–1680)