数学专业英语－（b）How to state a theorem?

yunjiang

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<  align=left>  定理叙述的格式，基本上与数学术语的定义一样，只不过在术语的定义中，“then”句有比较固定的格式，而定理的“then”句则随其结果而变吧了。<p></p></P>
<  align=left>1．某些定理可用简单句叙述。<p></p></P>
<P  align=left>The union of a finite number of closed sets is still a closed set.<p></p></P>
<P  align=left>The space <v:shapetype><v:stroke joinstyle="miter"></v:stroke><v:formulas><v:f eqn="if lineDrawn pixelLineWidth 0"></v:f><v:f eqn="sum @0 1 0"></v:f><v:f eqn="sum 0 0 @1"></v:f><v:f eqn="prod @2 1 2"></v:f><v:f eqn="prod @3 21600 pixelWidth"></v:f><v:f eqn="prod @3 21600 pixelHeight"></v:f><v:f eqn="sum @0 0 1"></v:f><v:f eqn="prod @6 1 2"></v:f><v:f eqn="prod @7 21600 pixelWidth"></v:f><v:f eqn="sum @8 21600 0"></v:f><v:f eqn="prod @7 21600 pixelHeight"></v:f><v:f eqn="sum @10 21600 0"></v:f></v:formulas><v:path connecttype="rect" gradientshapeok="t" extrusionok="f"></v:path><lock aspectratio="t" v:ext="edit"></lock></v:shapetype><v:shape><v:imagedata></v:imagedata></v:shape>(E,f) is complete.<p></p></P>
<P  align=left>2. 如果定理的结论是在一定前提下得到的，则可用下面形式：<p></p></P>
<P  align=left>   “Suppose…Then…”or“Let….Then…”<p></p></P>
<P  align=left>    Let f(x) be a continuous function defined on[a,b]. Then f(x) attains its maximum and minimum on [a,b].<p></p></P>
<P  align=left>    Suppose that f(z) is analytic in a simply connected domain D, then for any closed simple curve C lying within D, we have<p></p></P>
<P  align=left>                         <v:shape><v:imagedata></v:imagedata></v:shape> <p></p></P>
<P  align=left>3. 如果定理的结论在一定假设条件下成立，则可用下面的形式<p></p></P>
<P  align=left>   “If…, then…”<p></p></P>
<P  align=left>    If P(z) is a non-constant polynomial then there is a complex number c with P(c)=0<p></p></P>
<P  align=left>4. 如果定理的结论除了在一定条件下，还需在一定前提下才成立，这时可用如下形式<p></p></P>
<P  align=left>    “Let…. If…,then…”or<p></p></P>
<P  align=left>    “Suppose…. If…,then…”<p></p></P>
<P  align=left>     Let<v:shape> <v:imagedata></v:imagedata></v:shape>,<v:shape> <v:imagedata></v:imagedata></v:shape>,<v:shape> <v:imagedata></v:imagedata></v:shape>,<v:shape> <v:imagedata></v:imagedata></v:shape>be four distinct points. If all these four points lie on a circle, then the cross-ratio(<v:shape> <v:imagedata></v:imagedata></v:shape>,<v:shape> <v:imagedata></v:imagedata></v:shape>,<v:shape> <v:imagedata></v:imagedata></v:shape>,<v:shape> <v:imagedata></v:imagedata></v:shape>) is real.<p></p></P>
<P  align=left>5. 如果定理的结论在不同层次的几种条件下面成立，可用如下形式：<p></p></P>
<P  align=left>   “Let…, and assume….If…then…”<p></p></P>
<P  align=left>    Let f(x) be defined on open interval I, and assume that f(x) has a relative maximum or a relative minimum at an interior point c of I. If the derivative f’(c) exists, then f’(c)=0.<p></p></P>
<P  align=left> <p></p></P>