CSCI 246 Homework 6

• Due Date: Wednesday, March 30th no later than 11:59 p.m.
• Partner Information: You may complete this assignment individually or with exactly one classmate.
• Submission Instructions (working alone): Upload your solution, entitled YourFirstName-YourLastName to the BrightSpace Homework 6 Dropbox.
• Submission Instructions (working with exactly one classmate): Upload your solution, entitled YourFirstName-YourLastName-PartnerFirstName-PartnerLastName to the BrightSpace Homework 6 Dropbox. Note: If you work with a partner, only one person needs to submit a solution. If you both submit a solution, the submission that will be graded is the one from the partner whose last name comes alphabetically first.
• Deadline Reminder: Once the submission deadline passes, BrightSpace will no longer accept your submission and you will no longer be able to earn credit. Thus, if you are not able to fully complete the assignment, submit whatever you have before the deadline so that partial credit can be earned.

Assignment

Answer the following 13 questions in a clear and legible fashion in a single document. The document can be produced by software of your choice or handwritten.

1. Section 6.1. Let A = {x ∈ Z | x = 6a + 4 for some integer a} and B = {y ∈ Z | y = 18b - 2 for some integer b}. Prove or disprove A ⊆ B.
2. Section 6.1. Let B = {y ∈ Z | y = 18b - 2 for some integer b} and C = {z ∈ Z | z = 18c + 16 for some integer c}. Prove or disprove B ⊆ C.
3. Section 6.2. Use an element argument to prove that (A - B) ∩ (C - B) ⊆ (A ∩ C) - B for all sets A, B and C.
4. Section 6.3. Find a counterexample to show that the following statement is false for all sets A and B: (A ∪ B)^C = A^C ∪ B^C.
5. Section 6.3. Construct an algebraic proof that (A - B) ∪ (A ∩ B) = A for all sets A and B. Cite a property from Theorem 6.2.2. for every step.
6. Section 7.1. Show that function h: Q → Q, defined by the rule h(m/n) = m²/n for all integers m and n with n ≠ 0, is not well defined.
7. Section 7.2. Consider f(x) = (x + 1) / (x - 1) defined over the every real number except for x = 1. Is f one-to-one? Justify your answer.
8. Section 7.2. Let S be the set of all strings of 0's and 1's and define D: S → Z as follows: For every s ∈ S, D(S) = the number of 1's in s minus the number of 0's in s. Is D onto? Prove or give a counterexample.
9. Section 7.3. Define f: R → R and g: R → R by the formulas f(x) = x + 3 and g(x) = -x. Find f^-1, g^-1 and (g ∘ f)^-1.
10. Section 8.1. Let A = {0, 1, 3, 4, 5, 6} and define a relation V on A as follows: For every x,y ∈ A
    • x V y ↔ 5 | (x² - y²)
    Draw the directed graph of the relation defined by V.
11. Section 8.2. A is the absolute value relation on R: For all real numbers x and y, x A y ↔ |x| = |y|. Is A reflexive? Is A symmetric? Is A transitive? Justify your answers.
12. Section 8.2. Let T = {(0,2),(1,0),(2,3),(3,1)}. Find T^t, the transitive closure of T.
13. Section 8.3. Let P be the set of all points in the Cartesian plane except the origin. R is the relation defined on P as follows: For every p₁ and p₂ in P, p₁ R p₂ ↔ p₁ and p₂ lie on the same half-line emanating from the origin. (1) Construct a convincing argument that R is an equivalence relation. (2) How many distinct equivalence classes are there?

Grading - 50 points

• 30 points - 3 questions will be selected randomly to be graded carefully. Each of these questions is worth 10 points.
• 20 points - The other 10 questions are each worth 2 points. If you make a reasonable attempt, you will earn the 2 points.