# FOM Assignment 2

For this assignment, re-read sections 1.6-1.10, and read sections 2.1 – 2.3.  Some good examples to spend time understanding include the following: Example 1.8 (pg. 20), Example 1.11 (pg. 26), Example 2.4 (pg. 33).  Here are the problems to complete:

Problem 1. Exercises 1(a), 1(d), and 1(i) for Section 1.6

Problem 2. Exercises 3 and 6 for Section 1.6

Problem 3. Exercises 7, 8, and 14 for Section 1.7

Problem 4. Exercises 4(a), 9(a), and 10(a) for Section 1.8

Problem 5. Exercises 8 and 14 for Section 2.1

Problem 6. Exercises 2, 3, and 10 for Section 2.2

Problem 7. Exercises 5 and 8 for Section 2.3

Problem 8. What is the “well-ordering principle?”  On what page can you find it in our book?

Problem 9. What is the “division algorithm?” On what page can you find it in our book?

Problem 10.  Compose your own if-then statement, but be sure to make it a true one, and make it about some piece of popular culture or literature or art of which you are particularly fond.  For example, Harry Potter nerds might submit something like “If you are sorted into Gryffindor, then you are brave” whereas fans of the TV show Friends could write, “If you actually like Friends, you probably don’t have any.”

Problem 11.  For this problem suppose that $A$ and $B$ are non-empty sets (i.e. neither one of them equals $\emptyset$).

(a) Is it possible for $A \cup B$ to be empty?  If so, provide an example.  If not, explain why and restate this result as an if-then statement.

(b) Is it possible for $A \cap B$ to be empty?  If so, provide an example.  If not, explain why and restate this result as an if-then statement.

(c ) Is it possible for $A times B$ to be empty?  If so, provide an example.  If not, explain why and restate this result as an if-then statement.

Problem 12.  Consider the following statement: If $S$ is a set (with some universal set $U$), then $S \cap \overline{S} = \emptyset$.  Do you think this statement is true?  false?  Provide an explanation.

Problem 13.  Consider this statement: If $latex x \in \mathbb{R}$, then $x^x \in \mathbb{R}$.  Do you think this statement is true?  False?  Provide an explanation.

Problem 14.  What is an “open sentence?”

Problem 15.  Negate this statement: I like the television show “Friends” and I like Harry Potter.