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如何定义数学用语

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发表于 2009-2-5 10:44 |只看该作者 |倒序浏览
|招呼Ta 关注Ta
如何定义数学术语?
How to define a mathematical term?
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1.
& |# [+ s; y4 M: pSomething is defined as something.

$ @' P( e/ ]6 x( ]Something is called something.
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例如:
    The union of A and B is defined as the set of those elements which are in A, in B or in both.

# k8 J# i& ^  L) J1 wThe mapping ,is called a Mobius transformation.

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Something is defined to be something (or adjective)
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Something is said to be something (or adjective)

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例如:
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The difference A-B is defined to be the set of all elements of A which are not in B.
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A real number that cannot be expressed as the ratio of two integers is said to be an irrational number.
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Real numbers which are greater than zero are said to be positive.
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3.# M1 E4 z. \& g6 `7 e# q
We define something to be something.
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We call something to be something.
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例如:
We define the intersection of A and B to be the set of those elements common to both A and B.
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We call real numbers that are less than zero to be negative numbers.

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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:

. d+ f, S% v) D" r5 nLet
…, Then … is called …

. z8 B6 U) K: t+ h( uLet
…, Then … is said to be …
Let…, Then … is defined as …
Let…, Then … is defined to be …
例如:
Let x=(x1, x2, … xn) be an n-tuple of real numbers. Then the set of all such n-tuples is defined as the Euclidean n-space Rn .. [$ i3 ~9 ^' Q
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Let d(x,y) denote the distance between two points x and y of a set A. Then the number
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is called the diameter of A.
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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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If
…, then …is called …

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If
…, then …is said to be …

' F$ J# A. l  m6 b0 p* v3 x. G0 b  vIf
…, then …is defined as …

% x, R+ \+ O- G7 ^$ v, ]If
…, then … is defined to be …
例如:

1 q9 i0 f! G/ f+ |  kIf the number of rows of a matrix A equals the number of its columns, then A
9 D  N6 t) L0 {/ uis called a square matrix.

- F; \. g4 `$ d8 C$ xIf a function f is differentiable at every point of a domain D, then it is said to be analytic in D.
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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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Let(or Suppose)
…. If …, then … is called …
Let(or Suppose) …. If …, then … is said to be …
例如:
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Let f(z) be an analytic function defined on a domain D(
前提条件).If for every pair of points z1 and z2 in D with+ [" z+ }  a. V& n! q! b# U! N
z1≠z2 ,we have f(z1)≠f(z2) (直接条件),then f(z) is called a schlicht function or is said to be schlicht in D.

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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:

& ^; u2 K9 y! X/ O( P: z, ULet
…and suppose(or assume) …. If … then…is called…
例如:

  ~  M  g- T/ N/ W; t8 ~: l9 S4 qLet D be a domain and suppose that f(z) is analytic in D. If for every pair of points z1 and z2 in D with; {; N/ n+ ]1 a9 s, a# q
z1
≠z2 ,we have f(z1)≠f(z2),then f(z) is called a schlicht function .
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