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在数学论证中,我们常常用到举反例的方法。我认为,运用这种论证方法,有下面三个好处:一、举反例可以加深对定义、定理的条件的理解,澄清模糊认识。例如:棱台的定义——一棱锥被平行于底面的平面所截,夹在平面与底面间的部分叫棱台。有人提出了这样的问题:有两面平行,其它各面都是梯形的多面体是棱台吗?我们举一反例就可明白。如图一,这个多面体属于上面提到的一类,但根据定义它却并不是棱台。
In mathematical proofs, we often use the method of counter-examples. In my opinion, there are the following three advantages in applying this method of argumentation: First, to give negative examples can deepen the understanding of the conditions of definition and theorem and clarify the vague understanding. For example: The definition of the edge of a pyramid - a pyramid is parallel to the bottom surface of the plane is cut, the folder between the plane and the bottom of the part called the pyramids. Someone raised the question: Is there any parallelism on both sides, and is the polyhedron on the other side a trapezoid? As shown in Figure 1, this polyhedron belongs to the category mentioned above, but by definition it is not a platform.