(How to define a mathematical term?)! q. w* u4 r+ G% g
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Something is defined as something. % A% q& x) S; D7 E
Something is called something. 6 d. Y' A9 ?$ b3 w# S
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.
5 @& W: u* Y. [4 E8 m! g: f6 M. `The mapping ,is called a Mobius transformation.
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) z* Q# e, C# p g( \# Q: H6 _Something is defined to be something (or adjective)
7 @. j! D" w: w2 Z9 D) y+ DSomething is said to be something (or adjective)
; C$ J" Y6 K* e0 ~6 H/ h0 w n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.
! {. y. f( o, a* T0 T( ]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. 3.
1 s" ]$ k- k3 U+ UWe define something to be something.
2 O' W5 J: s: a/ ~* `9 U2 qWe 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.
: A+ O# ?5 S" A' d8 J) d8 g2 gWe call real numbers that are less than zero to be negative numbers.
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4.
; x! r6 v) B) y! O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:
" |$ o; V8 o# Q0 e' \7 q7 {& f- ~$ }Let…, Then … is called …
0 u( V# g6 G" D& e+ b* r" `/ u# `- J/ `Let…, Then … is said to be …
Let…, Then … is defined as … Let…, Then … is defined to be … 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 .
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1 u1 s/ W/ P; x& N- ^3 m- SLet d(x,y) denote the distance between two points x and y of a set A. Then the number
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:
) h' B8 \ Q% l0 Z6 |If …, then …is called …
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! {9 C0 v. F8 `# cIf …, then …is said to be …
1 I9 V- J1 l( OIf …, then …is defined as …
" r9 ^7 e8 e. u9 cIf …, then … is defined to be …
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If the number of rows of a matrix A equals the number of its columns, then A
$ x6 B/ {! Z; i. X9 D9 \is called a square matrix. + [% M( E+ q1 S. o
If a function f is differentiable at every point of a domain D, then it is said to be analytic in D. & g2 e, }5 l: T M0 _8 k' |9 r5 A. T
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: 5 Q, N9 [. j8 f9 A9 T3 |; A6 T
Let(or Suppose) …. If …, then … is called … Let(or Suppose) …. If …, then … is said to be …
4 m' Z2 K; N8 g7 L' c; N+ \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 with8 G8 ?5 v0 C U: s: n, {4 r8 [
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.
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:
% Q" O3 v0 I: z8 Y( DLet …and suppose(or assume) …. If … then…is called…
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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
7 F1 ^/ Z& V7 jz1≠z2 ,we have f(z1)≠f(z2),then f(z) is called a schlicht function . |