1 Homework 5 1. Enumerate the elements of the following relations from the set A of positive integers less than or equal to 10 to the set B of positive integers less than or equal to 30. (20 points, 2 parts- 10 points each) a. An element a of A is related to the element b of B if b = 3 × a b. An element a of A is related to the element b of B if b = 2 × a – 1 2. Determine the inverse of the following relations: (20 points, 2 parts- 10 points each) a. The > relation defined on the integers b. The = relation defined on the integers 3. Let A = {0, 1, 2, 3}. Define a relation R on A as follows: (20 points) R = {(0, 0), ((1, 1), (2, 2), (1, 2), (2, 1), (2, 3), (3, 2)}. Draw a directed graph for this relation and identify which of the following properties hold for this relation:  Reflexive  Symmetric  Transitive  Antisymmetric Explain why it has a property or give a counterexample. 4. Given the set A = {1, 2, 3} and the set S = {(x, y) | x and y in A}. Consider the relation ≤ defined on S as follows: ((x1, y1) ≤ (x2, y2) if x1 ≤ x2 and y1 ≤ y2. Draw the directed graph of this relation. Show that it is a partial order. Explain why it is not a total order. (20 points) 5. Consider the set S defined in problem 4 and the following relation = defined on S as follows: (x1, y1) = (x2, y2) if x1 + y1 = x2 + y2. Draw the directed graph of this relation. Show that it is an equivalence relation. List its equivalence classes. (20 points) Grading Rubric: Question Meets Does not Meet Question 1 20 points Innovative and correct method of solution. 0 points Solution not described, or not correct. 2 Calculations and supporting evidence are complete and correct for the problem. Solution is neat, well-organized and well-written. Calculations and supporting evidence are incorrect or not present. Solution is unorganized and poorly written. Question 2 20 points Innovative and correct method of solution. Calculations and supporting evidence are complete and correct for the problem. Solution is neat, well-organized and well-written. 0 points Solution not described, or not correct. Calculations and supporting evidence are incorrect or not present. Solution is unorganized and poorly written. Question 3 20 points Innovative and correct method of solution. Calculations and supporting evidence are complete and correct for the problem. Solution is neat, well-organized and well-written. 0 points Solution not described, or not correct. Calculations and supporting evidence are incorrect or not present. Solution is unorganized and poorly written. Question 4 20 points Innovative and correct method of solution. Calculations and supporting evidence are complete and correct for the problem. Solution is neat, well-organized and well-written. 0 points Solution not described, or not correct. Calculations and supporting evidence are incorrect or not present. Solution is unorganized and poorly written. Question 5 20 points Innovative and correct method of solution. Calculations and supporting evidence are complete and correct for the problem. Solution is neat, well-organized and well-written. 0 points Solution not described, or not correct. Calculations and supporting evidence are incorrect or not present. Solution is unorganized and poorly written.