数学专业英语第二25c.docx

数学专业英语第二25c数学专业英语书上2.5-C Sets of points in the plane We have already shown that there is a one-to-one correspondence between points in a plane and pairs of numbers (x,y) . Certain sets of points in the plane may be of special interest. For example , we may wish to examine the set of point comprising the circumference of a certain circle , or the set of points constituting the interior of a certain triangle . One may wonder if such sets of points may be succinctly described in compact mathematical notation. We may write (x,y)|y=2x to describe the set of ordered pairs (x,y) , or corresponding points , such that the ordinate is equal to twice the abscissas. In effect ,then, the vertical line in (1) is read “such that” . By “the graph of the set of ordered pairs” is meant the set of all points of the plane corresponding to the set of ordered pairs. The student will readily infer that the set of points constituting the graph lies on a straight line. Consistent the set (x,y)|y=x2 Consistent with our previous interpretation , this symbol represents the set of ordered pairs (x,y) such that the ordinate is equal to the square of the abscissa. Here ,the total graph comprises a simple recognizable geometrical figure , a curve known as a parabola. 数学专业英语书上2.5-C on the basis of these two example ,one may be tempted to believe that any ar-bitrarily drawn curve , which of course determines a set of points ordered pairs, could be described succinctly by a simple equation. Unfortunately ,this is not the case. For example , the broken line in figure 2-2-3 is one of such curves. Consider now the set (x,y)|y>2x to describe the set of points (x,y) whose ordinate is greater than twice its abscissa. In this case ,our set of point constitutes not a curve , but a region of the coordinate plane. 数学专业英语书上2.5-C 建立点在平面上 我们已经表明,在平面之间存在一一对应点和坐标(x,y)。点在平面上的某些设置可能的产生特殊的关系。例如,我们可能希望检查的点集组成的某一圈的周长,或点集构成一定的内部三角形。你可能想知道这样的集点可能在紧凑简洁地描述数学符号。 我们可以写 (x,y)|y=2x 描述的一组命令对(x,y),或对应点的纵坐标等于横坐标的两倍。实际上,(1)的垂直线是读“这样”。通过“图形的命令集对”是飞机的所有点的集合对应的命令集对。学生将很容易推断出这组点位于一条直线构成的图。 一致的集 (x,y)|y=x2 符合我们之前的解释,这个符号代表的一组命令对(x,y),纵坐标等于横坐标。在这里,总图包含一个简单的可识别的几何图形,一条曲线称为抛物线。 这两个例子的基础上,可能会相信任何一个任意绘制曲线,当然决定下令对一组点,可以简洁地描述一个简单的方程。不幸的是,情况并非如此。例如,折线图2-2-3是一个这样的曲线。 考虑现在的设置 (x,y)|y>2x 描述的集合点(x,y)的其横坐标纵坐标大于两次。在这种情况下,我们的点集构成不是一个曲线,但坐标平面的一个区域。