# Math 330: Number SystemsAnnouncements

### Section 4, Zaslavsky · Spring 2024

Main class page | Schedule and homework | Advice | Term Project | Syllabus

## Extra Problems

1. Implications.
1. P and Q are statements. What is the difference among "P if Q", "P is necessary for Q", and "P only if Q"?
2. Write the inverse, converse, and contrapositive of the following statement: "There can't be rain if there is no water."
3. Consider the statement "All apples are pears." Formulate it as an implication P ⇒ Q (say what P and Q are). Then state the contrapositive, converse, and inverse of your implication. Which of the four statements are true and which are false? Explain why.
4. This original question was a mistake: Consider the statement "No apples are pears." Formulate it as the negation of an implication R ⇒ S (say what R and S are). Then state the contrapositive, converse, and inverse of your implication. Simplify the negations of the original implication and the contrapositive, converse, and inverse.
Corrected question: Consider the statement "No apples are pears." Formulate it as an implication R ⇒ S (say what R and S are). Then state the contrapositive, converse, and inverse of your implication.
5. Compare the implications of parts c and d. Are they different? Why, or why not?
6. Consider the statement "Every natural number is positive." Do the same as part (c) for this statement.

2. Problem 4.3.1 (the "x+2 Problem"). This is just like the 3x+1 and x+1 problems except that xn+1 = xn + 2 for odd xn. Question: What is the long-term behavior of this sequence? How does it depend on the initial number m = x1?

3. Sets.
Project 5-A. Among the four sets A, B, F (Project 5.3) and
H = {3x−21: x ∈ Z and x > 7},
find all pairs such that (a) X = Y, (b) X ⊆ Y.
Advice: Check every pair of sets. Prove all your answers thoroughly, using the set definitions. That includes proving non-subsets.

4. Sets: 5.5.
In Project 5.5(iii), do you think m is supposed to be the same m for both Vm and Wm? What is the range of m supposed to be?
The problem is not stated clearly enough. (It's my fault; I wrote the problem for them.) The sets should be defined before the subproblem (i, ii, iii) statements. The range of m should appear within each subproblem statement. For instance, (iii) should read:
"Vm and Wm for a specified m ∈ Z."
See the discussion on set definition below.

5. Sets: 5.5+.
When reading or writing a set definition, pay attention to what is a variable inside the set definition and what is not a variable. As examples, how do the following pairs of sets differ? First, we define some sets:
• R = {my: y ∈ Z, m ∈ N, and my > 0}.
• S = {n: n ∈ N}.
• T = {n}.
• Um = {my: y ∈ Z and my > 0}.
• V = {2y: y ∈ Z and 2y > 0}.
• Wm = {my: y ∈ Z and y > 0}.

1. Now we state the pairs of sets to think about:
1. S, and T where n ∈ N.
2. R, and Um where m ∈ N.
3. Um and Wm, where m ∈ Z.
4. W2, and Wm with m = 2.
5. V, and Um with m = 2.

2. Now, find the simplest possible way of writing each of the sets in the preceding list.

6. Functions. See Chapter 5 on the advice page for definitions that are not in §5.4.
1. A certain function g: X → Y is defined by a table of values as follows: X = {cat, squirrel, sunflower, grass}, Y = {0, 1, 2, 3, 4, 5}, and
xg(x)
cat3
squirrel0
sunflower1
grass4

Here are some questions about g.
1. What is the domain, Dom(g)?
2. What is the codomain, Codom(g)?
3. What is the image, Im(g)?
4. Is g injective (one-to-one)?
5. Is g surjective (onto)?
6. Is g bijective (a one-to-one correspondence)?

2. Another function h: X → Y is defined by a table of values as follows, with the same X and Y, and
xh(x)
cat3
squirrel0
sunflower2
grass0

Here are some questions about h.
1. What is the domain, Dom(h)?
2. What is the codomain, Codom(h)?
3. What is the image, Im(h)?
4. Is h injective (one-to-one)?
5. Is h surjective (onto)?
6. Is h bijective (a one-to-one correspondence)?

## Test Coverage

### Midterm Test

The midterm covers Chapters 1 - 6 except §6.4. The format is open-book: you can have your textbook, but no notes and no electronic devices. Grading guidelines: (will be posted "soon")

``` 	A	B	C	D	F

```
Midterm test and solutions.

### Final Exam

The final exam covers all the material of the course. It will emphasize the parts we covered since the midterm, in Ch. 6, 8, 9, 10, 11, 13. The format is open-book (you can have your physical textbook, but no notes and no electronic devices).

Main class page | Schedule and homework | Advice | Term Project | Syllabus