Dedekind Cut - Representations

Representations

It is more symmetrical to use the (A,B) notation for Dedekind cuts, but each of A and B does determine the other. It can be a simplification, in terms of notation if nothing more, to concentrate on one 'half' — say, the lower one — and call any downward closed set A without greatest element a "Dedekind cut".

If the ordered set S is complete, then, for every Dedekind cut (A, B) of S, the set B must have a minimal element b, hence we must have that A is the interval ( −∞, b), and B the interval [b, +∞). In this case, we say that b is represented by the cut (A,B).

The important purpose of the Dedekind cut is to work with number sets that are not complete. The cut itself can represent a number not in the original collection of numbers (most often rational numbers). The cut can represent a number b, even though the numbers contained in the two sets A and B do not actually include the number b that their cut represents.

For example if A and B only contain rational numbers, they can still be cut at √2 by putting every negative rational number in A, along with every non-negative number whose square is less than 2; similarly B would contain every positive rational number whose square is greater than or equal to 2. Even though there is no rational value for √2, if the rational numbers are partitioned into A and B this way, the partition itself represents an irrational number.