Math 145 Test 2 Chapters 3 and 4 Study Sheet
Out of Class Essay
2 to 3 pages, typed, double spaced with a page of explanatory illustrations either
spread through the text or as a separate sheet. Title your illustrations so that it is
easy to know which one you are referring to in the text of your paper.
Choose one of the two following essays:
oThe Platonic Solids
Define and name the Platonic Solids
Explain how Euclid determined that there are not any more
Platonic Solids. Why was he sure there were no more?
Explain the concept of duality of Platonic Solids and how it is
related to the number of faces and the number of vertices.
oMultiple Infinities
Write a letter convincing a skeptical friend that infinity does
indeed come in different sizes.
With in your letter use good mathematical vocabulary, define the
idea of a one-two-one relationship, and describe at least different
three sizes of infinity.
In class portion
Your second test will consist of (2) questions from homework , and (3) essay questions
You will not know in advance which homework problems will appear on the test. Three
of these short essay questions will appear on the test. You may bring one 3x5 card with
notes.
Possible essay questions:
.
1. Given a finite set of cardinality 4, define its power set and state the cardinality of
the power set. If an additional element was added to the original set, how many
sets would be added to the power set? Justify your statement with logic.
Generalize your discussion to a rule about power sets.
2. Use an example to explain Cantor’s diagonal method of showing that that the
cardinality of the reals must be greater than the cardinality of the natural numbers.
3. Compare and contrast zero dimensions, one dimension, two dimensions, three
dimensions, and 4th dimension. Include the idea of a “slide.” Discussion either
the two and 3d safe problem or the knot problem as presented in the text.
4. Describe the method used to construct a golden rectangle. Based on your method
prove that the small rectangle within the original golden rectangle is also golden.
