EE 601 Week 5 Homework Instructions: Complete the following problems. Submit responses as a single PDF file in Black- board. For full credit, solutions must be justified and include supporting work when appropriate. 1. Let A be the set of people alive on earth. For each relation defined below, determine if it is an equivalence relation on A by investigating the three properties: reflexive, symmetric, and transitive. Be sure to justify your conclusions. If it is, describe the equivalence classes. If it is not, determine which properties of an equivalence relation fail. (a) xSy x and y were born in the same country. (b)cky + knows y. Hint: Make a definition for how knows is interpreted in the response. 2. Let A = {1, 2, 3, 4, 5, 6}. Form a partition of A using {1, 2}, {3}, and {4, 5, 6}. These are the equivalence classes for the equivalence relation E on A. (a) Draw the digraph of E. (b) Compute the 0-1 matrix of E using the natural order of A for the rows and columns. Com- ment on any structure that you observe. 3. Determine if 1883 is prime. Be sure to justify your conclusions. 4. Explain why it is incorrect to write the expression 3 12 = 4. Hint: It has to do with the difference between the division operation and divides relation. 5. Determine all integers that 0 divides. Hint: It is true that division by 0 is undefined which is a statement about the division operation. This problem asks a different question. It asks about the divides relation, namely find all integers n such that On is true. The definition of the divides relation will be handy here. 6. Arrange the numbers -46, -27, -8, 18, 37, 61, 91, so that they are in the order 0, 1, 2,3, 4, 5, 6 modulo 7. Be sure to justify your conclusions