- Announcements:
- Return HW3
- Homework on Section 2.2, problems 16-24, due today.
- Problems for Monday, 2/23: Section 3.2, pp. 156-161, Mindscapes #6-8, 14, 15, 20
- New Assignments:
- We have an exam next Wednesday: I want you to get started reviewing!
- Your exam will go through our "Infinity" unit. That means we will cover four units on the exam:
- Fun and Games
- Estimation
- Number Systems
- Infinity
- New Assignments:
- Your assignment for section 3.4 will not be handed
in: but it will help prepare you for the
exam. Problems: p. 186, #5, 6, 7, 8.
- For Monday, read section 4.7: The Fourth Dimension
- A poem by Robert Graves: Warning to Children
- Section 3.2: Comparing the Infinite
- The natural numbers are infinite in number (the size of the set is
larger than any natural number).
- Are there more evens or more natural numbers?
-
- World inside a world....
- Are there more primes or more natural numbers?
- Are there more Fibonaccis or more natural numbers?
-
- 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 take another look at these curious problems (pp. 157):
- 13 - squaring off
- 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
- Section 3.4: stratosphere of infinities
- The Power set
- set of all subsets of a set
- Power sets lead to bigger sets
- this is true even in the case of infinite sets!
- A proof (following the idea of Dodge Ball) that there is an
infinite set too big to be contained in "Motel Infinity".
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