数学专业英语－(a) How to define a mathematical term?

yunjiang · 发表于 2004-5-6 10:05:26

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> 
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> 数学术语的定义和数学定理的叙述，其基本格式可归纳为似“if…then…”的格式，其他的格式一般地说可视为这一格式的延伸或变形。
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> 如果一定语短语或定语从句，以界定被定义的词，所得定义表面上看虽不是“If……then……”的句型，而实际上是用“定语部分”代替了“If”句，因此我们可以把“定语部分”写成If句，从而又回到“If……then……”的句型。
 至于下面将要叙述的“Let…if…then”，“Let and assume…, If…then…”等句型，其实质也是基本句型“If……then……”的延伸。
 有时，在定义或定理中，需要附加说明某些成份，我们还可在“if…then…”句中插入如“where…”等的句子，加以延伸（见后面例子）。
 总之，绝大部分（如果不是全部的话）数学术语的定义和定理的叙述均可采用本附录中各种格式之。
 
 
<H2 align=center><A>（a</A>）How to define a mathematical term?</H2>
 
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is defined as
is called</TD></TR></TABLE>
1. Something something
 
 
 The union of A and B is defined as the set of those elements which are in A, in B or in both. 
 The mapping <v:shapetype><v:stroke joinstyle="miter"></v:stroke><v:formulas><v:f eqn="if lineDrawn pixelLineWidth 0"></v:f><v:f eqn="sum @0 1 0"></v:f><v:f eqn="sum 0 0 @1"></v:f><v:f eqn="prod @2 1 2"></v:f><v:f eqn="prod @3 21600 pixelWidth"></v:f><v:f eqn="prod @3 21600 pixelHeight"></v:f><v:f eqn="sum @0 0 1"></v:f><v:f eqn="prod @6 1 2"></v:f><v:f eqn="prod @7 21600 pixelWidth"></v:f><v:f eqn="sum @8 21600 0"></v:f><v:f eqn="prod @7 21600 pixelHeight"></v:f><v:f eqn="sum @10 21600 0"></v:f></v:formulas><v:path connecttype="rect" gradientshapeok="t" extrusionok="f"></v:path><lock aspectratio="t" v:ext="edit"></lock></v:shapetype><v:shape><v:imagedata></v:imagedata></v:shape>, ad-bc<v:shape> <v:imagedata></v:imagedata></v:shape>0, is called a Mobius transformation. 
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is defined to be
is said to be </TD></TR></TABLE>
2. Something something(or adjective)
 
 
 The difference A-B is defined to be the set of all elements of A which are not in B.
 A real number that cannot be expressed as the ratio of two integers is said to be an irrational number.
 Real numbers which are greater than zero are said to be positive.
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define
call</TD></TR></TABLE>
 3. We something to be something.
 
 
 We define the intersection of A and B to be the set of those elements common to both A and B.
 We call real numbers that are less than zero (to be) negative numbers.
 4. 如果在定义某一术语之前，需要事先交代某些东西（前提），可用如下形式：
 
 
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is called
is said to be
is defined as
is defined to be</TD></TR></TABLE>
 Let…, then…
 
 
 
 
 Let x=(<v:shape> <v:imagedata></v:imagedata></v:shape>) be an n-tuple of real numbers. Then the set of all such n-tuples is defined as the Euclidean n-space R.
 Let d(x,y) denote the distance between two points x and y of a set A. Then the number
 D= <v:shape><v:imagedata></v:imagedata></v:shape>
 is called the diameter of A.
5．如果被定义术语，需要满足某些条件，则可用如下形式：
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<TR >
<TD vAlign=top width=120>
is called
is said to be
is defined as
is defined to be</TD></TR></TABLE>
 If…, then… 
 
 
 
 
 If the number of rows of a matrix A equals the number of its columns, then A is called a square matrix.
 If a function f is differentiable at every point of a domain D, then it is said to be analytic in D.
6.如果需要说明被定义术语应在什么前提下，满足什么条件，则可用下面形式：
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is called
is said to be</DIV></TD></TR></TABLE><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1026"></v:textbox></v:shape>
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Let
Suppose</DIV></TD></TR></TABLE> …. If…then… …
 
 
 Let f(z) be an analytic function defined on a domain D (前提条件). If for every pair or points<v:shape> <v:imagedata></v:imagedata></v:shape>, and <v:shape><v:imagedata></v:imagedata></v:shape> in D with <v:shape><v:imagedata></v:imagedata></v:shape><v:shape><v:imagedata></v:imagedata></v:shape><v:shape><v:imagedata></v:imagedata></v:shape>, we have f(<v:shape> <v:imagedata></v:imagedata></v:shape>)<v:shape> <v:imagedata></v:imagedata></v:shape>f(<v:shape> <v:imagedata></v:imagedata></v:shape>) (直接条件)，then f(z) is called a schlicht function or is said to be schlicht in D.

yunjiang · 发表于 2004-5-6 10:05:38

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0cm 0cm 0pt 18pt; TEXT-INDENT: -18pt; TEXT-ALIGN: left; mso-char-indent-count: -1.5; mso-char-indent-size: 12.0pt" align=left>7. 如果被定义术语需要满足几个条件（大前提，小前提，直接条件）则可用如下形式：<

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0cm 0cm 0pt">supposeassume</DIV></TD></TR></TABLE> Let…and …. If…then…is called… Let D be a domain and suppose that f(z) is analytic in D. If for every pair of points <v:shape><v:imagedata></v:imagedata></v:shape> and <v:shape><v:imagedata></v:imagedata></v:shape> in D with<v:shape> <v:imagedata></v:imagedata></v:shape><v:shape><v:imagedata></v:imagedata></v:shape><v:shape><v:imagedata></v:imagedata></v:shape>, we have f(<v:shape> <v:imagedata></v:imagedata></v:shape>)<v:shape> <v:imagedata></v:imagedata></v:shape>f(<v:shape> <v:imagedata></v:imagedata></v:shape>), then f(z) is called a schlicht function. Notes:(a) 一种形式往往可写成另一种形式。Let{<v:shape> <v:imagedata></v:imagedata></v:shape>}be a sequence of sets. If<v:shape> <v:imagedata></v:imagedata></v:shape>for all n, then{<v:shape> <v:imagedata></v:imagedata></v:shape>}is called an ascending or a non-decreasing sequence.我们可用一定语短语来代替“If”句，使其变为“Let……then”句Let{<v:shape> <v:imagedata></v:imagedata></v:shape>}be a sequence of sets with<v:shape> <v:imagedata></v:imagedata></v:shape>for all n, then{<v:shape> <v:imagedata></v:imagedata></v:shape>}is called an ascending or a non-decreasing sequence.(b) 注意“Let”，“suppose”（“assume”），“if”的使用次序，一般来说，前面的可用后面的替换，但后面的用前面的替换就不好了，如上面句子可改写为： Suppose{<v:shape> <v:imagedata></v:imagedata></v:shape>}is a sequence of sets. If<v:shape> <v:imagedata></v:imagedata></v:shape>, then{<v:shape> <v:imagedata></v:imagedata></v:shape>}is called an ascending sequence. Let{<v:shape> <v:imagedata></v:imagedata></v:shape>}be a sequence of sets and suppose that<v:shape> <v:imagedata></v:imagedata></v:shape>then{<v:shape> <v:imagedata></v:imagedata></v:shape>}is called an ascending sequence. 但下面的句子是错误的（至少是不好的句子）； If{<v:shape> <v:imagedata></v:imagedata></v:shape>}is a sequence of sets, and let<v:shape> <v:imagedata></v:imagedata></v:shape>, then{<v:shape> <v:imagedata></v:imagedata></v:shape>}is called an ascending sequence.(c) 在定义一些术语后，往往需要用符号来表达，或者需要对句中某些成份作附加说明，这时我们需要把定义句扩充，扩充的办法是在定义的原有结构中，插入一个由连接词引导的句子，这类连接词或短语经常是“and”，“where”，“in this (that) case”…请参看PARTIA第一课注1和第二课注4、5、6。If every element of a set A also belongs to another set B, then A is said to be the subset of B, and we write <v:shape><v:imagedata></v:imagedata></v:shape> A real number is said to be a rational if it can be expressed as the ratio of two integers, where the denominator is not zero.(d) 在定义中，“if”句是关键句，且往往比较复杂，要特别注意在一些定义中，“if”句又有它自己的表达格式，读者对这类句子的结构也要掌握，下面我们以函数极限定义中的“if”句的结构作为例子加以说明：If for every<v:shape> <v:imagedata></v:imagedata></v:shape>＞0, there is (there exists) a <v:shape><v:imagedata></v:imagedata></v:shape>＞0, such that <v:shape><v:imagedata></v:imagedata></v:shape>＜<v:shape> <v:imagedata></v:imagedata></v:shape>whenever 0＜<v:shape> <v:imagedata></v:imagedata></v:shape>＜<v:shape> <v:imagedata></v:imagedata></v:shape>, then we say f(x) has a limit A at the point a. 上面是函数极限的定义，其中的“if”句是它的典型结构，凡与极限相关的概念，如连续，收敛，一致连续，一致收敛等定义均有类似结构。例： A sequence of functions {<v:shape> <v:imagedata></v:imagedata></v:shape>} is said to have the Cauchy property uniformly on a set E if for any <v:shape><v:imagedata></v:imagedata></v:shape>＞0, there is an N such that<v:shape> <v:imagedata></v:imagedata></v:shape>＜<v:shape> <v:imagedata></v:imagedata></v:shape>whenever n,m＞N. 当然，极限定义还有其他表达形式但基本结构是一样的，只不过对句中某些部分用等价的语法结构互作替换而已。 下面是函数极限定义中“if”句的另一些表达式，读者可把这些句子和原来的句子作比较。 If, given any<v:shape> <v:imagedata></v:imagedata></v:shape>＞0, there exists a <v:shape><v:imagedata></v:imagedata></v:shape>＞0, such that<v:shape> <v:imagedata></v:imagedata></v:shape>＜<v:shape> <v:imagedata></v:imagedata></v:shape>whenever (if,for) 0＜<v:shape> <v:imagedata></v:imagedata></v:shape>＜<v:shape> <v:imagedata></v:imagedata></v:shape>,… If, corresponding to any <v:shape><v:imagedata></v:imagedata></v:shape>＞0, a <v:shape><v:imagedata></v:imagedata></v:shape>＞0 can be found such that<v:shape> <v:imagedata></v:imagedata></v:shape>＜<v:shape> <v:imagedata></v:imagedata></v:shape>whenever 0＜<v:shape> <v:imagedata></v:imagedata></v:shape>＜<v:shape> <v:imagedata></v:imagedata></v:shape>,… If, for every <v:shape><v:imagedata></v:imagedata></v:shape>＞0,there is a <v:shape><v:imagedata></v:imagedata></v:shape>＞0, such that 0＜<v:shape> <v:imagedata></v:imagedata></v:shape>＜<v:shape> <v:imagedata></v:imagedata></v:shape>implies <v:shape><v:imagedata></v:imagedata></v:shape>＜<v:shape> <v:imagedata></v:imagedata></v:shape>.

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