By Bak A. (ed.)

**Read Online or Download Algebraic K-Theory, Number Theory, Geometry and Analysis: Proceedings of July 26-30, 1982 PDF**

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**Additional info for Algebraic K-Theory, Number Theory, Geometry and Analysis: Proceedings of July 26-30, 1982**

**Sample text**

Then there exist lines L1 , L2 , and L3 such that P2 \ P3 D L1 , P1 \ P3 D L2 , and P1 \ P2 D L3 . Furthermore, one and only one of the following statements is true: (1) L1 \ L2 D L1 \ L3 D L2 \ L3 D ;, (2) L1 D L2 D L3 , and (3) L1 \ L2 \ L3 is a singleton. Fig. 9, showing alternative (1) at left, alternative (2) in the middle, and alternative (3) on the right. Proof. 2. 4, P2 \ P3 is a line L1 , P1 \ P3 is a line L2 , and P1 \ P2 is a line L3 . P1 \ P3 // D P1 \ P2 \ P3 Â L3 . Hence L1 \ L2 Â L3 .

7. Let E be a plane. There exists a point P such that P … E. Proof. 5 there exist points A, B, C, and D which are noncoplanar. If fA; B; C; Dg were a subset of E, then A, B, C, and D would be coplanar. Hence at least one member of fA; B; C; Dg does not belong to E, proving the theorem. 8. Let S and T be distinct planes whose intersection is the line L, and let P be a member of L; then there exist lines M and N such that M Â S, N Â T , M ¤ L, N ¤ L, and M \ N D fPg. If M and N are any two lines satisfying these conditions, then there is exactly one plane E such that M [ N Â E.

11. Every plane contains (at least) three distinct lines. 12. Space contains (at least) six distinct lines. 13 . If L is a line contained in a plane E, then there exists a point A belonging to E but not belonging to L. 14. If P is a point in a plane E, then there is a line L such that P 2 L and L Â E. 15. If a plane E has (exactly) three points, then each line contained in E has (exactly) two points. 16. If a plane E has exactly four points, and if all of the lines contained in E have the same number of points, then each line contained in E has (exactly) two points.