Math 320 – Spring 2022: Problem Sets

Welcome to Math 320 – Chaos and Fractals! I will post assignments and announcements here throughout the semester.  Check back frequently. Below are links to some resources we will be using in the course.

1. Read the course syllabus below.
2. Purchase a copy of Maple (linked to in the syllabus) and install it on your laptop, or be sure you can access it via remote.scranton.edu or a public computer in the math lab.
3. Prove the Alternate Or- theorem: $$(P\text{ or }Q) \text{ and }\sim\!P\Rightarrow Q$$ Use the semiformal proof style, with one statement per line, indented subproofs, and reasons for every line that needs them, and only use the rules of inference for Natural Deduction from the notes, not any famous theorems about propositions. You don’t need to use line numbers or cite premises, however. The precedence of logical operators is specified on page 14 of the lecture notes for Math 299.
4. Write up and hand in your answer to problems #1,#2, and #3 in the Logic and Proofs section of the Homework Problems pdf linked to below. For #3 use the same proof style as in the Alternate Or- problem above.

Write up and hand in your answer to the following problems from the Homework Problems pdf linked to below. You can use semi-formal proofs but no theorems of logic – just the rules of inference given in the notes. The problems are numbered by the indicated section of the homework pdf.

1. Logic and Proofs #4a,b,c,d, #5
2. Sets, Functions, and Sequences #1, #2

Write up and hand in your answer to the following problems from the Homework Problems pdf linked to below. You can use semiformal proofs and whatever theorems of logic you like – but no theorems about functions from set theory. These problems are in the revised lecture notes document below.

• #2.3 #2.4, #2.7, #2.8, #2.9, #2.10 (we proved one direction in class so you don’t need to do that half), #2.11, #2.12

Due Thursday, February 10

Write up and hand in your answer to the following problems from the the revised lecture notes document below.

• #2.13, #2.14, #2.15

Due Tuesday, February 15

Write up and hand in your answer to the following problems from the the revised lecture notes document below.

• #3.3, #3.4, #3.5, #3.6, #3.7, #3.8

Due Thursday, February 17

Write up and hand in your answer to the following problems from the the revised lecture notes document below.

• #3.9, #3.10, #3.11, #3.12

Due Tuesday, February 22

Write up and hand in your answer to the following problems from the the revised lecture notes document below.

• #3.13, #3.14, #3.15, #3.16

Due Thursday, February 24

Write up and hand in your answer to the following problems from the the revised lecture notes document below.

• #3.17, #3.18

Due Tuesday, Mar 1

Write up and hand in your answer to the following problems from the the revised lecture notes document below.

• #3.20, #3.21, 3.23, 3.24

Due Thursday, Mar 3

Write up and hand in your answer to the following problems from the the revised lecture notes document below.

• #3.22, #3.25, #3.26

Due Thursday, Mar 3

Write up and hand in your answer to the following problems from the the revised lecture notes document below.

• #4.1, 4.2, 4.5, 4.6, 4.7, 4.8(b), 4.10

Thursday, Mar 10

Study for the midterm exam. It will be in-class on paper. Bring an approved highlighter marker with you, writing utensils, and a cheap non-programmable calculator with a square root button.

Due Thursday, Mar 3

Write up and hand in your answer to the following problems from the the revised lecture notes document below.

• #4.12, 4.13, 4.14, 4.11 (do this one last after you finish the first three)

Due Thursday, Mar 24

Write up and hand in your answer to the following problems from the the revised lecture notes document below.

• #5.3, 5.5, 5.6

Due Tuesday, Mar 29

Write up and hand in your answer to the following problems from the the revised lecture notes document below.

• #5.7, 6.1, 6.2, 6.3

Due Tuesday, Mar 29

1. Dr. Monks wrote out the proof of Theorem 13.7 in the appendix of the revised lecture notes below, but he forgot to supply the reasons! Help him out by supplying the reasons for all of the lines in that proof.
2. Let $f\colon \mathbb{R}^2\to\mathbb{R}^2$ by $f(x,y)=(x/3+1,y/5+1)$ for all $x,y\in\mathbb{R}$.
   1. Prove that $f$ is a contraction map.
   2. Determine (with proof) a contraction factor for $f$.
   3. Find the fixed point of $f$.
   4. Compute and plot the first ten iterations of the $f$-orbit of $(10,10)$. Does it seem to be converging to the fixed point you found?

Due Tuesday, April 5

Write up and hand in your answer to the following problems from the the revised lecture notes document below. You can use earlier parts of problems to prove later parts, but not the other way around (unless you prove them first… no circular arguments).

• #7.1, 7.2, 7.3, 7.4

Due Tuesday, April 5

Write up and hand in your answer to the following problems from the the revised lecture notes document below. You can use earlier parts of problems to prove later parts, but not the other way around (unless you prove them first… no circular arguments).

• #7.5
• #7.6 (use the definition of scalar multiplication, matrix multiplication, and matrix addition, given in the notes and a big transitive chain)
• #7.7 (in the Maple chaos package the syntax to convert a Maple affine expression into a function and apply it to a point is, e.g. Map(Affine(1/2,1/2,0,0,0,0))(2,4), which will return $[1,2]$)
• #7.9
• #7.13

Due Tuesday, April 5

Write up and hand in your answer to the following problems.

• #7.14
• #7.16
• Play “Guess My IFS” online linked to below, like we did in class. Solve three different levels (your choice), and hand in a screenshot showing the levels you solved and the message saying your answer is correct. Pick your favorite three.

Due Tuesday, April 19

Write up and hand in your answers to the following problems.

• #8.17
• #8.18
• #8.10

Due Thursday, April 21

Write up and hand in your answers to the following problems.

• Play Three Rounds of Guess My Address – with three different fractals. Take a screen shot showing your high scores for all three and hand them in with your assignment. Naturally try to score as high as you can!
• #9.3

Due Tuesday, April 26

Write up and hand in your answers to the following problems.

• #9.1
• #9.2
• #9.4

Due Thursday, April 28

Write up and hand in your answers to the following problems.

• #10.1
• #10.2
• #10.3

Due Tuesday, May 3

Write up and hand in your answers to the following problems.

• #10.4
• #10.5
• #10.8
• #10.9

Due Thursday, May 5

Write up and hand in your answers to the following problems.

• #10.6
• #10.7

Due Tuesday, May 10

Write up and hand in your answers to the following problems.

• #11.3
• #11.5
• #11.6
• #11.4

Due Thursday, May 12

Write up and hand in your answers to the following problems.

• #10.10
• #11.7