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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 |正序浏览

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如何定义数学术语?

（How to define a mathematical term?）. t! M$ Y, u% ^5 l+ {. E+ K

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1.
Something is defined as something.

Something is called something.

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.

The mapping ,is called a Mobius transformation.

2.
Something is defined to be something (or adjective)

Something is said to be something (or adjective)

例如：

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.

3.
We define something to be something.

We call something to be something.

例如：

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：

Let…, Then … is called …

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 .

Let 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：

If …, then …is called …

If …, then …is said to be …

If …, then …is defined as …

If …, then … is defined to be …

例如：

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：

Let(or Suppose) …. If …, then … is called …

Let(or Suppose) …. If …, then … is said to be …

例如：

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
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：

Let …and suppose(or assume) …. If … then…is called…

例如：

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
z1≠z2 ，we have f(z1)≠f(z2)，then f(z) is called a schlicht function .

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