Wk Marun

Instructions: Complete the following problems. Submit responses as a single PDF file in Blackboard. 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) xKy x 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. Comment 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 0|n 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.