区分定理(Theorem)、引理(Lemma)、推论(Corollary)等概念
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发布时间:2022-11-07 02:48
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时间:2023-11-22 09:54
名词解释
Theorem:就是定理,比较重要的,简写是 Thm。
Lemma:小小的定理,通常是为了证明後面的定理,如果证明的篇幅很长时,可能会把证明拆成几个部分来叙述,虽然篇幅可能变多,但脉络却很清楚。
Corollary:推论。由定理立即可推知的结果。
Property:性质,结果虽然值得一记,却没定理来的深刻。
Proposition:有人翻译为「命题」, 有些作者喜欢用,大概也可以算是比较简单的定理的一种称呼。
Claim:证明时先叙述一个结果,再作证明。看的人比较轻松。
Note:通常只是一个注解。
Remark:涉及一些结论,比较起来 "Note" 比较像说明, "remark" 则常是非正式的定理。
首先、 定义 和 公理 是任何理论的基础,定*决了概念的范畴,公理使得理论能够被人的理性所接受。
其次、 定理 和 命题 就是在定义和公理的基础上通过理性的加工使得理论的再延伸,我认为它们的区别主要在于,定理的理论高度比命题高些,定理主要是描述各定义(范畴)间的逻辑关系,命题一般描述的是某种对应关系(非范畴性的)。而 推论 就是某一定理的附属品,是该定理的简单应用。
最后、 引理 就是在证明某一定理时所必须用到的其它定理。而在一般情况下,就像前面所提到的定理的证明是依赖于定义和公理的。
1.引理和定理应该是根据文章目的不同而区分的,同样的论点在这篇文章可以是引理,在那篇文章可以是定理。
2.如果为了说明一个问题进行论证,但是在论证前需要证明若干个小问题,那么这些若干个小问题的结论就是引理,而这个问题的论证将会需要引用到前面的引理,该问题的结论就是定理。
3.引理是为定理作准备的。文章中的定理才是需要说明的主要问题或者目的。
就如doppler 说的,
"Theorem" 本身是一个大 result
"Lemma" 是 prove “Theorem“ 之前用的一个 result
"Corollary" 是可以从 "Theorem" 里直接 dece/prove 出来的 result
" Proposition" 是一个还无法大到变成 "Theorem" 的一个 result (当作小 theorem
)
(1) Definition(定义)------a precise and unambiguous description of the meaning of a mathematical term. It characterizes the meaning of a word by giving all the properties and only those properties that must be true.
(2) Theorem(定理)----a mathematical statement that is proved using rigorous mathemat-ical reasoning. In a mathematical paper, the term theorem is often reserved for the most important results.
(3) Lemma(引理)----a minor result whose sole purpose is to help in proving a theorem. It is a stepping stone on the path to proving a theorem. Very occasionally lemmas can take on a life of their own (Zorn's lemma, Urysohn's lemma, Burnside's lemma,Sperner's lemma).
(4) Corollary(推论)-----a result in which the (usually short) proof relies heavily on a given theorem (we often say that \this is a corollary of Theorem A").
(5) Proposition(命题)-----a proved and often interesting result, but generally less important than a theorem.
(6) Conjecture(推测,猜想)----a statement that is unproved, but is believed to be true (Collatz conjecture, Goldbach conjecture, twin prime conjecture).
(7) Claim(断言)-----an assertion that is then proved. It is often used like an informal lemma.
(8) Axiom/Postulate------(公理/假定)a statement that is assumed to be true without proof. These are the basic building blocks from which all theorems are proved (Eu-clid's ve postulates, Zermelo-Frankel axioms, Peano axioms).
(9) Identity(恒等式)-----a mathematical expression giving the equality of two (often variable) quantities (trigonometric identities, Euler's identity).
(10) Paradox(悖论)----a statement that can be shown, using a given set of axioms and de nitions, to be both true and false. Paradoxes are often used to show the inconsistencies in a awed theory (Russell's paradox). The term paradox is often used informally to describe a surprising or counterintuitive result that follows from a given set of rules(Banach-Tarski paradox, Alabama paradox, Gabriel's horn).
http://blog.sina.com.cn/s/blog_a0e53bf70101jwv1.html