- Announcements:
- UCLA group discovers massive prime number
- Sorry, the exams aren't graded yet. I should have them finished by Wednesday.
- There's one idea that we'll want to examine before we move onto
our unit on Dimension: that there's an infinity bigger than that of the
natural numbers. For next time, read section 3.4: a stratosphere of
infinities. The fundamental notion is that of the power set:
make sure that you understand this concept before next time, to make it
easier to understand the material.
- Note: there is a homework set due Friday....
- Section 3.2: Comparing the Infinite
- The natural numbers are infinite in number (the size of the set is
larger than any natural number).
-
- What is Cardinality?
- Two sets have the same cardinality if there
is a one-to-one correspondence between them.
Intuitively: they're the same size.
-
- 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!
- Let's look at these curious problems (pp. 157):
- 14 - counting cubes
- 16 - Hotel Cardinality
- 17 - Hotel Cardinality (cont.)
- 18 - Hotel Cardinality and the Infinite Life Insurance Co.
- Hotel Cardinality and the Limitless Bus Barn Co.
- The ping-pong ball conundrum
- The ping-pong ball conundrum is related to other paradoxical
ideas, such as Zeno's paradoxes
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