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有如下四个命题：

①平面α和平面β垂直的充要条件是平面α内至少有一条直线与平面β垂直；

②平面α和平面β平行的一个必要不充分条件是α内有无数条直线与平面β平行；

③直线a与平面α平行的一个充分不必要条件是平面α内有一条直线与直线a平行；

④两条直线平行是这两条直线在一个平面内的射影互相平行的既不充分也不必要条件．

其中正确的序号是______．

答案

①平面α和平面β垂直的充要条件是平面α内至少有一条直线与平面β垂直，是一个正确命题，由定理过一个平面的垂线的平面与这个平面垂直；

②平面α和平面β平行的一个必要不充分条件是α内有无数条直线与平面β平行，正确命题，两平面平行，可得出一个平面中有无数条直线与另一个平面平行，反之，一个平面中有无数条直线与另一个平面平行，推不出两平面平行；

③直线a与平面α平行的一个充分不必要条件是平面α内有一条直线与直线a平行，错误命题，此是一个充要条件；

④两条直线平行是这两条直线在一个平面内的射影互相平行的既不充分也不必要条件，正确命题，两条直线平行时，在同一个平面内的投影可以是两个点，不能得射影平行，如果两个直线在平面内的投影平行，两直线的位置关系可能是异面．

故答案为：①②④

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It is clear that we are rapidly becoming a global culture. New forms of information technology, intercontinental travel, and the complex nature of economics are all elements in such a change. Japan, an ancient (1) culture, now dominates the West economically while its own traditional (2) is rapidly becoming (3) in the process.
It does not take a great deal of analysis to see the vast interchanges of cultural influences in the world today. The thirst for (4) fashions, popular music, and the other elements of consumer culture (5) great inroads in non-Western lands while the West seeks out everything from the religious traditions of the Ease (6) the art and music these countries produce. In the field of (7) , to cite one conspicuous example, contemporary readers seek books (8) those of the West. Latin American authors, as well as those from Africa, Asia, and India, now find appreciative audiences.
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