Math 402, Assignment 2, Spring 2008
Due Wed. January 30 (at the beginning of class)
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Look at the list of axioms
which we have been using in class.
For each part of the five axioms, decide whether it is true or
false for a cylinder. Explain how you know that it is true
and/or exactly when and how it fails. Make sure that you consider
each possible type of line and every possible configuration of points.
Write down the best
possible variation of the axiom which is true.
Draw pictures where appropriate.
Note: To test the plane separation axiom, try actually cutting
the cylinder.
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a) On the plane, consider
two distinct lines l and m.
Do they always intersect in exactly one point? If not,
what are the possibilities? Explain each case as completely
as possible.
b) Answer the same question for the sphere.
c) Recall that for the plane, all
of the axioms are true, but for the sphere, some are false. Use the axioms
to explain why your answers to a) and b) are different. Make sure
that you consider each case and refer to the specific part(s) of the
specific axiom(s) that you
are using.
d) Turn this into a formal proof that your answer in a) is correct.
-
a) On the plane, consider
two distinct points p and q.
Is there always exactly one segment between them? If not,
what are the possibilities? Explain each case as completely
as possible.
b) Answer the same question for the sphere.
c) Answer the same question for the cylinder.
d)
Use the axioms
to explain why your answers to a), b), and c) are different. Make sure
that you consider each case and refer to the specific part(s) of
the specific axiom(s) that you
are using.