Math 360, Dr. Vaden-Goad, Spring 2003, practice test 1, name __________________
Think carefully, show all work
Fold test lengthwise with essay on the outside
1. Write a brief essay (1 page: introduction, development and conclusion) in which you
discuss the uses of models when assessing a set of axioms.
2-6 will be basic recall and short answer. You are responsible for all the words used
between page 50 and page 65, even if the definition occurs elsewhere. For instance, the
word “parallel” occurs between page 50 and page 65, but the definition is on page 19.
You are responsible for that definition.
examples:
Define parallel
State the elliptic parallel property.
What is an affine plane?
What is the projective completion of an affine geometry?
What is a projective plane?
7-10 will be proofs. They will be of a form where you must first decide whether the
statement is true or false in the stated context and then prove that you are correct. If the
statement is independent, you may give models demonstrating independence without
proving your statements about the models. In the examples that follow, the things in
parentheses are there to help you understand how to solve the example. They won’t be
on the test.
examples:
Prove or Disprove: For all incidence geometries, there are at least four distinct
points. (This one is false, and all you need to do is cite the example worked out in the
book of a three point incidence geometry.)
Prove or Disprove: In all affine planes, there are at least four distinct points. (This
one is true and so you will have to somehow do a universal proof – I suggest RAA so you
will have a given affine plane with fewer than four points: Look for a contradiction)
Prove or Disprove: In all projective planes, there are at least four distinct points.
(You get the picture – take any statement you understand and start changing it in small,
controlled ways. Now ask whether it is provable, disprovable or independent. For this
series of questions, you can start increasing the number of points as you add conditions to
the hypothesis --- but what restrictions require how many points? Don’t expect to be able
to answer every question you can ask, but do a lot of them so that you can see the
diversity of possible questions and get a sense of how to begin to attack each of them.
My task in writing the test is to pick questions challenging enough to be interesting, but
still reasonable. Subject to those restrictions, I’ll pick a range of questions. Unless
otherwise specified, you can use anything between page 50 and 65.)
Prove or Disprove: I2 is independent of I1, I3 and the Euclidean Parallel Property.
(That is, among geometries satisfying I1, I2 and the Euclidean Parallel Property, there is
one which satisfies I2 and there is one satisfying ~I2.)
Prove, Disprove or show independent: (Given I1, I2 and I3) There exist three
concurrent lines. (Note here that I1, I2 and I3 do not fully specify the context of the
statement, thus we must allow for the possibility that the statement is independent (it is.))
Prove, Disprove or Show independent: If M is a hyperbolic incidence geometry,
there are three non-concurrent lines. (Proof: Let M be a hyperbolic incidence geometry.
Since M satisfies I1 and I3, Lemma 2 provides three non-concurrent lines.)
(Play with restrictions on the type of geometry for the hypothesis and various numbers of
distinct points, distinct lines and concurrent lines for the conclusion. Play with assuming
various of our axioms and one or more of our propositions or properties as hypothesis
and various other propositions, axioms or properties as conclusion. You will find it very
helpful to keep our various models in mind as well as the actual proofs of the
propositions.)