代做IS 203 Exercise Set 6代做迭代

IS 203

Exercise Set 6

Exercise Sets should be completed through group work, but each student will submit their own solutions. The following problems must be submitted through the assignment page in Canvas by 11:59 pm on Sunday, October 6, 2024.

Exercise sets are distributed as Word files so students have an easier time using the assignment. To properly see some symbols, the document must be opened in Word. Students who prefer to print out the exercise set and write their answers (or write answers on a tablet directly onto a document or pdf) should first insert extra lines between questions and problems.

Items highlighted yellow must be complete before class on Monday.
Items highlighted blue must be complete before class on Wednesday.

Items highlighted pink must be complete before class on Friday.

Items with no highlighting are suggestions of when to complete work in order to complete the exercise set on time.

Part 1: Working With Sets

1. What is the cartesian product of:

a. A = {1, 2}
B = {1, 5}

b. A = {1, 1, 3}
B = {2, 1, 7}

Use the following sets to answer the below questions.
A = {1, 2, 3, 4} B = {1, 2, 3, 4, 5, 6}
C = {1, 3, 5, 7, 9} D = {2, 4, 6, 8, 10}

1. Circle the statements that are true. If a statement is false, explain why it is false.
A = B
B ⊆ A
B ⊃ A
A ⊂ B

2. A ∪ B =

3. A ∩ B =

4. C ∩ D =

5. A ∪ B ∪ C ∪ D =

6. A ∩ B ∩ C =

Part 2: The Limits of Logic

→ Task: Watch lecture video 21.

Part 3: Sorites Paradox

Can Sorites Paradox be demonstrated or proven using Predicate Logic? Please explain your answer.

Part 4: Matrices

→Task: Read or skim chapter 2.4 of the Rosen text. Watch lecture video 22.

Special note: Lecture video 22 and in-class work are optional based on the given circumstances:

Watch and attend class if:

· You do not know what a matrix is.

· You have performed matrix calculations before but do not remember them well.

· You are unfamiliar with how to answer any problem in Part 5.

· You are unfamiliar with parts of chapter 2.4.

You may skip Monday’s class if:

· You have taken the course, “Linear Algebra.”

· You are well acquainted with matrix calculations before and are familiar with all properties of matrices discussed in the text.

· You are familiar with the material in chapter 2.4.

Part 5: Matrix Calculations

Use the below matrices to perform. the given operations. Some operations might not be permitted. In such cases, please state why.

1. A + A

2. C + D

3. AC

4. BA

5. AB

6. CE

7. F²

8. B²

9. F³

10. F⁰

11. E⁸

12. I_n for A

13. I_n for B

14. I_n for C

15. A^t

16. C^t

17. Which of the above matrices is symmetric? For each symmetric matrix, explain the significance of being symmetric.

Part 6: Functions

→Task: Watch lecture videos 23 & 24. Read Chapters 2.3 – 2.3.2 from the Rosen text. Some of the material in the text is not included in the video. The intent is for students to learn to read mathematical material and interpret it.

Part 7: Functions

Use the below functions for problems 1-4.

f(x) = 2x g(x) = x – 1 h(x) = x^x k(x) = x²

1. f(g(x))

a. Compose and simplify the function.

b. Solve for -2, -1, 0, 1, 2.

c. Is the composed function one-to-one for ℤ? Why or why not?

2. g(f(x))

a. Compose and simplify the function.

b. Solve for -2, -1, 0, 1, 2.

c. Is the composed function one-to-one for ℤ? Why or why not?

3. g(h(x))

a. Compose and simplify the function.

b. Solve for -2, -1, 0, 1, 2.

c. Is the composed function one-to-one for ℤ? Why or why not?

4. k(f(x))

a. Compose and simplify the function.

b. Solve for -2, -1, 0, 1, 2.

c. Is the composed function one-to-one for ℤ⁺? Why or why not?

Part 8: Sequences & Series

→Task: Watch lecture videos 25 and 26. Skim Chapter 2.4 of the Discrete math text.

Comprehension Assessment 3

The comprehension assessment is due at the same time as the exercise set and can be submitted with the exercise set as a single document.

Total Grade Value: 100 points

Grading:

· All proofs must be written using the forms learned in class.

· No more than 50% of a point value for a part will be deducted no matter how many errors.

· 0% of the point value for a question, problem or section will be awarded if no work is shown or if a response is not attempted.

· Detailed rubrics for each part are included on the Canvas page for the Comprehension Assessment.

Part 1: 20 points

In English, explain whether propositional or predicate logic is required to prove the below argument and why. Please do not construct a proof.
1. Humans are mortal
2. Socrates is human
⸫ Socrates is mortal

Part 2: 20 points

Prove: ꓯx(¬Fx ν Gx) → ¬ꓱx¬(Fx → Gx)

Part 3: 20 points

Below is one of the two Quantifier Negation rules learned in class. In English, explain why the two statements on each side of the biconditional are equivalent to each other. It might help to translate the quantifiers into English.

¬ ∀xFx ↔ ∃x¬ Fx

Part 4: 20 points

The below matrix will be used for all problems in Part 3.

1. A^t =

2. A⁰ =

3. A⁴ =

4. A x I₃ =

Part 5: 20 points
Use the below sets to answer questions for Parts 3.

A = {1, 3, 5, 8} B = {0, 1, 1, 2, 3, 5, 8, 13}

C = {2, 4, 6, 10, 12} D = {1, 2, 3}

1. ꓯx{x ∈ A ↔ x ∈ D}
Is the above logical statement true? Please explain your answer.

2. A ∪ D =

3. A ∩ B ∩ D =

4. ℤ⁺ - D =

Extra Credit: (2 points if attempted, 4 points if correct)

Is a negative hypothesis associated with “evidence of absence” or “absence of evidence?” Why?