A projective plane is a set of points and subsetscalled lines that satisify the

Question

A projective plane is a set of points and subsetscalled lines that satisify the following four axioms: 1. Any two distinct points lie on a unique line 2. Any two lines meet in at least one point 3. Every line contains at least three points 4. There exists three noncolinear points Given these four axioms prove, (a) Every projective plane has at least seven points, andthese exists a model of a projective plane having exactlyseven points. (b) The projective plane of seven points is unique up toisomorphism. A projective plane is a set of points and subsetscalled lines that satisify the following four axioms: 1. Any two distinct points lie on a unique line 2. Any two lines meet in at least one point 3. Every line contains at least three points 4. There exists three noncolinear points Given these four axioms prove, (a) Every projective plane has at least seven points, andthese exists a model of a projective plane having exactlyseven points. (b) The projective plane of seven points is unique up toisomorphism.

Explanation / Answer

keeping the 4th , 3rd axioms in view, there can be 9 points onthe three non collinear lines.
but by the 2nd axiom , we can easily see that line 1, line 2have a common point and line 2 , line 3 have a common point. so, omitting the repeatition of 2 points from the possible 9 ,we get that there are atleast 7 points on a projectiveplane.

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