Finite Geometry Assignment
1. For Finite Geometry Axiom System 1, figure out and draw a model
that fits the axioms and has 5 things in it.
Finite
Geometry Axiom system 1:
There exists a set of objects called
Things
There exists another set of objects called Bunches. Each
Bunch is a set that includes some Things.
If there is a Thing and a Bunch then either the Thing is in the Bunch
or the Thing is not in the Bunch.
- For every two distinct Things there is exactly one Bunch that
contains both of them
- Any time there is a Bunch, and a Thing that is not in that Bunch,
then there is another Bunch that contains the thing, and does not
intersect with the first Bunch.
- There exist at least 4 Things where no three of those Things are
in the same Bunch.
2. For Finite Geometry Axiom System 2, explain why the model
below doesn't satisfy these axioms (what axiom does it not satisfy and
why)?
Finite
Geometry Axiom system 2:
There exists a set of
objects
called Things.
There exists a collection of objects called Bunches, where each line is
a subset of the set of Things.
(Given a Thing and a Bunch, the Thing is either in the Bunch or it is
not in the Bunch)
- For every two distinct Things there is exactly one Bunch that
contains both of them
- Every pair of Bunches intersect, and the intersection
of
any two
distinct Bunches is a single Thing.
- There exists a set of four Things, no three of which
are in
the
same Bunch.
Model:
