集合

出自維基百科,自由嘅百科全書
跳去: 定向搵嘢
呢篇文介紹數學上所謂嘅「初等/模素/平凡/intuitive/naive」集論;Naive set theory有更詳細嘅料;Axiomatic set theory講現代嚴格嘅公理化集合。

(或集合)係一拃互相唔同嘅嘢,而被當做一樣嘢來研究。集係數學中最基本嘅概念之一。集嘅研究,集論,而家重發展緊。集論喺19 世紀尾至被發現/發明,到而家,喺數學中已經無所不在;集合論可以用來做數學嘅一種基礎,由佢可以導出差唔多全部數學。數學教育中,集論嘅基本原素(如Venn 圖)細細個就可以學;深入啲嘅嘢喺大學至會教。

哲學中,集通常被當做抽象嘅嘢en:abstract objects[1][2] [3] [4] 佢物質(physical)上嘅tokens就係,例如:檯面三隻杯夾埋一齊被叫做 "嗰啲杯"("the cups"),或者黑板上構成開閂大括夾埋其間嘅任何符號形狀嘅粉筆線。但係,mathematical realism嘅支持者,包括 Penelope Maddy 就有論話集合係實在嘅嘢en:concrete objects)。

The intersection of two sets is made up of the objects contained in both sets, shown in a Venn diagram.

定義[編輯]

康托de:Georg Cantor)喺佢嘅Beiträge zur Begründung der transfiniten Mengenlehre開頭咁樣定義「集」:[5]

By a "set" we mean any collection M into a whole of definite, distinct objects m (which are called the "elements" of M) of our perception [Anschauung] or of our thought.

一集嘅元素,又叫「成員」,係乜都得:字母,第啲集,諸餘此類;集通常都用大草拉丁字母來代表。「 A 等於 B 」 即係話佢哋嘅成員完全一樣(i.e., every member of A is also a member of B and vice versa)。

唔同「multiset」,一集入面每一成員都係單丁嘅;無兩樣嘢係一模一樣(identical)嘅。 任何集嘅運算(operations)都保守住呢樣性質;而且集嘅定義中唔計成員嘅次序,唔似序列en:sequence)同tuple

[編輯]

  1. Rosen, Gideon, "Abstract Objects", The Stanford Encyclopedia of Philosophy (Spring 2006 Edition), Edward N. Zalta (編), [1]
  2. Partee, Barbara Hall; ter Meulen, Alice G. B.; Mathematical Methods in Linguistics, [2]
  3. Brown, James Cooke; Sets and Multiples, [3]
  4. Goldstein, Laurence; "Representation and geometrical methods of problem solving", Forms of Representation: an Interdisciplinary Theme for Cognitive Science, Donald Peterson, ed.,. Exeter: Intellect Books, 1996. [4]
  5. (Dauben, p. 170 嘅英譯)