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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?
- F: b5 U% G- i! T. k# N. w

) ?+ M* F' `3 v# d; z

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Something is defined as something.
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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.

( J% f8 j1 |6 K( V' r$ s2 `The mapping ,is called a Mobius transformation.

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Something is defined to be something (or adjective)

$ n2 \3 V5 k0 E" T6 jSomething is said to be something (or adjective)
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例如:

: z6 \' j$ R) s4 MThe 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.
, J- O/ ]1 b" |2 JWe define something to be something.

4 q; c" x  m" N( j" R1 X& nWe 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:

( \. i# l% }6 T  S4 |1 t& vLet
…, Then … is called …
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Let
…, 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 .5 k) [8 \3 Z, \4 {! }* [. U6 ]2 ^% V
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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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# C3 [$ w$ D, R7 U: @If
…, then …is said to be …

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…, then …is defined as …

' @2 |( I% X$ d3 p8 h1 ^If
…, then … is defined to be …
例如:

: Y' d5 c1 ^0 ]' l; U7 CIf the number of rows of a matrix A equals the number of its columns, then A9 K) C! r9 v, ?2 A0 {# S! z, m' v
is called a square matrix.
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If 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:

- d7 z1 c* T* F, gLet(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
/ @& `% S+ b8 T7 m; n/ Sz1≠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:

$ X! O% d; M2 r& F$ x, RLet
…and suppose(or assume) …. If … then…is called…
例如:

! F7 C+ y7 W: n3 I: C4 `3 r+ _' q! @Let 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
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≠z2 ,we have f(z1)≠f(z2),then f(z) is called a schlicht function .
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