3 Let S be the following statement in the language of incide

3. Let S be the following statement in the language of incidence geometry: \"If l and m are any two distinct lines, then there exists a point P that does not lie eitherl or m.\" Show that S cannot be proved from the axioms of incidence geometry Show that S holds in every projective geometry. (Note: (a) and (b) implies that S is independent of incidence axioms.) Use (b) to show that in any finite projective geometry, all the lines have the same number of points lying on them (Hint: Given any two lines l and m, define a function : {1} {m} such that is bijective, which implies that the number of points on l equals the number of points on m.) (a) (b) (c)

Solution

As per the axioms, it mentions that there is atleast one point that doesn\'t lie on a line but it doesn\'t say that it lies on other lounge or not. This it can\'t be inferred that a given line doesn\'t lie on any two given lines