##
80-110 The Nature of Mathematical Reasoning

**Summer 2000**
*Dirk Schlimm*

# Homework No. 15

Monday, August 7, 2000

Due Thursday, August 10, 2000

1. **1-1 Functions.** (2 points)

Take a set A={alligator, bear, chimpanzee} and a set B={apple, banana,
cherry}. Write down two functions from A to B, one of which is 1-1 and
one which is not.

2. **Cardinality.**
(4 points)

Prove: The cardinality of the even numbers is the same as the
cardinality of the natural numbers.

3. **The Non-Denumerability of the
Continuum.** (4 points)

Read Handouts #22 and #23.

- Explain the proof of the theorem:
*The interval of all real
numbers between 0 and 1 is not denumerable*, on pages 259-261, so
that your fellow students who have not read the handout can understand
it.
- State in one or two sentences the key idea behind this proof.
- State in one or two sentences what the importance of this proof is.

4. **Preparation for final exam.** (5
points)

Write a paragraph or two answering both of the following questions:

- What is the difference between mathematical reasoning and other forms of reasoning, e.g., scientific,
historical?
- What is a proof?

(You probably have noticed that these are exactly the same questions as
in Homework #1. By now, considering the amount of material you have
learned in this course, your answers should be a bit different than
what you've written 5 weeks ago. Answering this question gives you the
opportunity to review the material and reflect about what
we have discussed in this course.)
5. ** (Optional). Set theory.**

If you are
interested to know more about Russell's Paradox and modern set theory,
read FOL, pages 216-224 (Sections
8.5-8.8). Attempt the following problem:

- Page 220, Problem 26. (5 extra points)