Dedekind cut

In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the rational numbers into two non-empty sets A and B, such that all elements of A are less than all elements of B, and A contains no greatest element. The set B may or may not have a smallest element among the rationals. If B has a smallest element among the rationals, the cut corresponds to that rational. Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the 'gap' between A and B. In other words, A contains every rational number less than the cut, and B contains every rational number greater than or equal to the cut. An irrational cut is equated to an irrational number which is in neither set. Every real number, rational or not, is equated to one and only one cut of rationals.A Dedekind cut is a partition of the rationals Q {displaystyle mathbb {Q} }   into two subsets A and B such thatIt 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'.Regard one Dedekind cut (A, B) as less than another Dedekind cut (C, D) (of the same superset) if A is a proper subset of C. Equivalently, if D is a proper subset of B, the cut (A, B) is again less than (C, D). In this way, set inclusion can be used to represent the ordering of numbers, and all other relations (greater than, less than or equal to, equal to, and so on) can be similarly created from set relations.A typical Dedekind cut of the rational numbers Q {displaystyle mathbb {Q} }   is given by the partition ( A , B ) {displaystyle (A,B)}   withA construction similar to Dedekind cuts is used for the construction of surreal numbers.