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# MODERN GEOMETRY MATH 532

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This 1 page Class Notes was uploaded by Cassidy Grimes on Monday October 26, 2015. The Class Notes belongs to MATH 532 at University of South Carolina - Columbia taught by Ma Filaseta in Fall. Since its upload, it has received 9 views. For similar materials see /class/229525/math-532-university-of-south-carolina-columbia in Mathematics (M) at University of South Carolina - Columbia.

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Date Created: 10/26/15

MATH 5327361 LECTURE 4 1 Finish Previous Notes 2 Homework Assign further problems from Homework 1 Quiz 012709 Tuesday 3 Finite Projective Planes Given a positive integer 71 called the order Axiom P1 There exist at least 4 points no 3 of which are collinear Axiom P2 Axiom P3 Axiom P4 There exists at least 1 line with exactly 71 1 distinct points on it Given 2 distinct points there is exactly 1 line that they both lie on Given 2 distinct lines there is at least 1 point on both of them 4 Prove the duals of the above axioms in the following order Dual of Axiom P4 Given 2 distinct points there exists at least 1 line passing through both of them Dual of Axiom P3 Given 2 distinct lines there exists exactly 1 point on both of them Dual of Axiom P1 There exist at least 4 distinct lines no 3 of which are concurrent Dual of Axiom P2 There exists at least 1 point with exactly 71 1 lines passing through it Comments Give handout of the proof of the dual of Axiom P1 before proving it Before the proof of the dual of Axiom P2 mention that the following will be shown Lemma If Z is a line with exactly 71 1 points on it in a nite projective plane of order n and A is a point not on Z then there exist exactly 71 1 lines passing through A 5 Main Steps for Lemma 0 LetP1 PnH be the points on Z 0 Let Zj be the line through A and Pi use Axiom P3 0 No Zj is Z A is not on Z 0 The Zj are distinct use Axiom P3 hence at least 71 1 lines go thru A 0 Let Z be a line through A 0 Then some Pi is on Z use Axiom P4 0 Deduce Z Zj by Axiom P3 so no more than n 1 lines go thru A 6 Main Steps for Dual of P2 0 Let Z be a line with exactly 71 1 points on it use Axiom P1 0 Let A be a pont not on Z use Axiom P1 0 Apply the lemma

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