An extract on #visualambassadors
The official Austrian dictionary, das sterreichische Wrterbuch, prescribes grammatical and spelling rules defining the official language. Austrian delegates participated in the international working group that drafted the German spelling reform of 1996several conferences leading up to the reform were hosted in Vienna at the invitation of the Austrian federal governmentand adopted it as a signatory, along with Germany, Switzerland, and Liechtenstein, of an international memorandum of understanding (Wiener Absichtserklrung) signed in Vienna in 1996. The "sharp s" () is used in Austria, as in Germany.
Because of the German language's pluricentric nature, German dialects in Austria should not be confused with the variety of Standard German spoken by most Austrians, which is distinct from that of Germany or Switzerland. Distinctions in vocabulary persist, for example, in culinary terms, where communication with Germans is frequently difficult, and administrative and legal language, which is due to Austria's exclusion from the development of a German nation-state in the late 19th century and its manifold particular traditions. A comprehensive collection of Austrian-German legal, administrative and economic terms is offered in Markhardt, Heidemarie: Wrterbuch der sterreichischen Rechts-, Wirtschafts- und Verwaltungsterminologie (Peter Lang, 2006).
There are many other equivalent statements of the axiom of choice. These are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it.
One variation avoids the use of choice functions by, in effect, replacing each choice function with its range.
Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.
This guarantees for any partition of a set X the existence of a subset C of X containing exactly one element from each part of the partition.
Another equivalent axiom only considers collections X that are essentially powersets of other sets:
For any set A, the power set of A (with the empty set removed) has a choice function.
Authors who use this formulation often speak of the choice function on A, but be advised that this is a slightly different notion of choice function. Its domain is the powerset of A (with the empty set removed), and so makes sense for any set A, whereas with the definition used elsewhere in this article, the domain of a choice function on a collection of sets is that collection, and so only makes sense for sets of sets. With this alternate notion of choice function, the axiom of choice can be compactly stated as
Every set has a choice function.
which is equivalent to
For any set A there is a function f such that for any non-empty subset B of A, f(B) lies in B.
The negation of the axiom can thus be expressed as:
There is a set A such that for all functions f (on the set of non-empty subsets of A), there is a B such that f(B) does not lie in B.