- Your fractal assignment is returned today today, and I give you "the winners".
A few words about this assignment:
- When you construct a fractal, you must explain the simple rule for your fractal (not the "simple rule" that there is a simple rule that you apply over and over, do it again, do it again, etc.). I need to know your rule!
- I've decided that, rather than a homework assignment on infinity,
I'll give you a little quiz on Wednesday about infinity, that will
count as a homework assignment. That way you'll know the sort of
questions that I might ask on the exam, RE infinity. Is that okay?
That way you can also get immediate feedback.
- We're in the reviewing mode of the course. The next three days
will not involve new material, per se.
- Your "cheat sheet" for the final will be two 8.5x11 (inches!)
sheets of paper, formed together into a mobius band. They
must be formed into a mobius band, or you can't use them.
- It is time for course evaluations: please visit eval.nku.edu to give your feedback (and also to make sure that you will get access to your grades earlier).
- Wrapping up Infinity! A really big idea....
. What we've learned so far about infinity:
- Infinity is NaN: Not a Number
- Infinity is NaS: Not a Size, either! The reason is that there
is more than one size of infinity! This is the most
important thing I want you to remember. (This fact will allow
you to win on the playground!;)
- The natural numbers are infinite in number (
).
- Cardinality allows us to distinguish between sizes
of infinity:
- Two sets have the same cardinality if there is a one-to-one correspondence between them.
- A sensible rule you can be sure of: a subset is never
bigger in size than the set itself.
And if the sets are finite, the proper subset is always smaller....
but if the set is infinite, we may actually be able to throw away elements of a set and not change the size of the set! - The evens, the odds, the rational numbers -- all are the
same size as the natural numbers.
- But Cantor showed that the real numbers are larger than
the naturals -- because we couldn't find a one-to-one
correspondence between them.
In fact, we proved that it's impossible to have a one-to-one correspondence (by contradiction: we assumed that there was such a correspondence, then showed that that led to an impossibility).
- The power set: showers of infinities!
The power set of a set
is the set of all subsets of
If
elements, then the power set of
, has
elements.
There are infinitely many sizes of infinity. It turns out that the power set of a set is always of larger cardinality than the set itself, even for infinite sets. So
- We ended on a really strange note: The ping-pong ball conundrum (which is related to other paradoxical ideas, such as Zeno's paradoxes). While we threw in ping pong balls 10 times faster than we drew them out, in the end, after one minute, there were no ping pong balls left in the barrel! Who can explain that?
- Infinity is NaN: Not a Number
- Now let's take a look at our home page, and the topics we've
covered:
topics
Who has a question about any of the areas we've covered so far?
- If there is any time left, we'll take a mathematical look at a "fun and game": tic tac toe....