...Consider the system [S, L, P], where S contains exactly four points A, B, C, and D, the lines are the sets with exactly two points, and the planes are sets with exactly three points. This "space" is illustrates by the following figure:
Here it should be remembered that A, B, C, and D are the only points that count. Verify that all the incidence postulates hold in this system.
I0)All lines and planes are sets of points.
I1) Given any two different points, there is exactly one line containing them.
I2) Given any three different noncollinear points, there is exactly one plane containing them.
I3) If two points lie in a plane, then the line containing them lies in the plane.
I4) If two planes interesect, then their intersection is a line.
I5) Every line contains at least two points. S contains at least three noncollinear points. Every plane contains at least three noncollinear points. And S contains at least four noncoplanar points.
This is what I have thought about doing so far to prove each incidence postulate but I'm not sure if it is right or not:
I0) This holds becuase the hypothesis states the lines are sets with exactly two points.
I1) This holds because the hypothesis states that the planes have exactly three points in them. no.. you have to prove that for any 2 distinct points, there is a unique line that contains both points. the hypothesis only says that for any line, it contains exactly two points, but not the other way around.. that is what you are going to prove..
I2) use the same argument as I1.
I3) This holds because if A lies in a plane and B lies in the same plane, then the line joining them must lie in the same plane. a consequence of I1 and I2.
I4) This holds because there is an exact point, D in S, which is both a line and a plane. first, you have to name two planes.. take their intersection. their intersection must contain 2 points (exactly). by I1, you conclude that it is line.
I5) The first statement hold because it is given in the hypothesis. The second statement holds because we are given A, B, C, D and no more than three are lying on the same line. The last one is satisfied because no four of the points are lying on the same plane. the last statement can be proven by enumeration. take the 4 points, then show that if you take 3 points at a time, the last point must not be on the plane where the 3 points are located..
This is my first class having to prove things, so please bear with me.
Thanks for your help!!