Teori order: Perbedaan antara revisi

 

Notasi 0 sering dijumpai pada elemen terkecil, meskipun tidak melibatkan bilangan apapun. Namun, dalam tatanan suatu himpunan bilangan, notasi ini tidak tepat dan bahkan menimbulkan kerancuan, karena bilangan 0 tidak selalu yang terkecil. Contohnya adalah pada tatanan divisibilitas |, di mana 1 adalah elemen terkecil karena bilangan itu membangi semua bilangan yang lain. Sebaliknya, bilangan 0 merupakan bilangan yang dapat dibagi oleh semua bilangan lain. Jadi bilangan 0 merupakan '''elemen terbesar''' dari tatanan tersebut. Istilah lain untuk "terkecil" dan "terbesar" adalah "terendah" ("terbawah", "paling dasar"; ''bottom'') dan "tertinggi" ("teratas"; ''top'') dan juga "nol" (''zero'') dan "unit" ("satuan").

 

<!--Least and [[greatest element]]s may fail to exist, as the example of the real numbers shows. But if they exist, they are always unique. In contrast, consider the divisibility relation | on the set {2,3,4,5,6}. Although this set has neither top nor bottom, the elements 2, 3, and 5 have no elements below them, while 4, 5, and 6 have none above. Such elements are called '''minimal''' and '''maximal''', respectively. Formally, an element ''m'' is [[minimal element|minimal]] if:

 

Lower bounds again are defined by inverting the order. For example, -5 is a lower bound of the natural numbers as a subset of the integers. Given a set of sets, an upper bound for these sets under the subset ordering is given by their [[union (set theory)|union]]. In fact, this upper bound is quite special: it is the smallest set that contains all of the sets. Hence, we have found the '''[[least upper bound]]''' of a set of sets. This concept is also called '''supremum''' or '''join''', and for a set ''S'' one writes sup(''S'') or v''S'' for its least upper bound. Conversely, the '''[[greatest lower bound]]''' is known as '''[[infimum]]''' or '''meet''' and denoted inf(''S'') or ^''S''. These concepts play an important role in many applications of order theory. For two elements ''x'' and ''y'', one also writes ''x''&nbsp;v&nbsp;''y'' and ''x''&nbsp;^&nbsp;''y'' for sup({''x'',''y''}) and inf({''x'',''y''}), respectively. <!-- Using Wikipedia's [[meta:MediaWiki User's Guide: Editing mathematical formulae|TeX markup]], one can also write <math>\vee</math> and <math>\wedge</math>, as well as the larger symbols <math>\bigvee</math> and <math>\bigwedge</math>. Note however, that all of these symbols may fail to match the font size of the standard text and should therefore preferably be used in extra lines. The rendering of ∨ for v and ∧ for ^ is not supported by many of today's [[web browser]]s across all platforms and therefore avoided here.-->

For another example, consider again the relation | on natural numbers. The least upper bound of two numbers is the smallest number that is divided by both of them, i.e. the [[least common multiple]] of the numbers. Greatest lower bounds in turn are given by the [[greatest common divisor]].