数学专业英语(Doc版).Word2
-
数学专业英语－
(a) How to define a
mathematical term?
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。
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<
/p>
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
„
,
< br>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。
（
a
）
How
to
define
a
mathematical
term?
is defined as
is
called
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
,
ad-bc
0,
is
called
a
Mobius
transformation.
is defined to be
is said to be
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.
define
call
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.
3.
We
something
to
be
something.
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：
is
called
is said to be
is defined as
is defined to be
Let
d(x,y)
denote
the
distance
between
two
points
x
and
y
of
a
set
A.
Then
the
number
D=
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：
is called
is
said to be
is defined as
is defined to be
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：
is
called
is said to
be
Let
Suppose
„
.
If
„
then
„
„
Let
f(z)
be
an
analytic
function
defined
on
a
domain
D
(
前提条件
).
If
for
every
pair
or
points
,
and
in
D
with
,
we
have
f(
)
f(
)
(
直接条件
)
，
then
f(z)
is
called
a
schlicht
function
or
is
said
to
be
schlicht
i
n
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：
If
„
,
then
„
If
the
number
of
rows
of
a
matrix
A
equals
the
number
of
its
columns,
then
A
is
called
a
square
matrix.
Let
„
,
then
„
Let x=(
)
be
an
n-tuple
of
real
numbers.
Then
the
set
of
all
such
n-tuples
is
defin
ed
as
the
Euclidean
n-space
R.
suppose
assume
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
and
in
D
with
,
we
have
f(
)
f(
),
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{
}be
a
sequence
of
sets.
If
for
all
n,
then{
}is
called
an
ascending
or
a
non-decreasing
sequence.
我们可用一定语短语来代替“
If<
/p>
”句，使其变为“
Let
„„
then
”句
Let{
}be
a
sequence
of
sets
with
for
all
n,
then{
}is
called
an
ascending
or
a
non-decreasing
sequence.
(b)
注意“
Let
”，“
suppose
”（“
assume
”），“
if<
/p>
”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{
}is
a
sequence
of
sets.
If
,
then{
}is
called
an
ascending
sequence.
Let{
}be
a
sequence
of
sets
and
suppose
that
then{
}is
called
an
ascending
sequence.
但下面的句子是错误的（至少是不好的句子）；
If{
}is
a
sequence
of
sets,
and
let
,
then{
}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
< br>是“
and
”，“
where<
/p>
”，“
in
this
(that)
case
”„请参看<
/p>
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
w
e
write
A
real
number
is
said
to
be
a
rational
if
it
can
be
expressed
as
the
ratio
of
two
integers,
where
the
de
nominator
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
＞
0, there
is (there exists) a
＞
0, such
that
＜
whenever
0
＜
＜
,
then we say f
(x) has a limit A at the
point a.