Unit 21 Exercises
1. Which of the following relations on {1, 2, 3, 4} are equivalence relations? Determine the properties of an equivalence relation that the others lack.
a) {(1, 1), (2, 2), (3, 3), (4, 4)}
b) {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}
c) {(1, 1), (1, 2), (1, 4), (2, 2), (2, 4), (3, 3), (4, 1), (4, 2), (4, 4)}
2. Suppose that A is a nonempty set, and f is a function that has A as its domain. Let R be the relation on A consisting of all ordered pairs (x, y) where f(x) = f(y).
3. Show that propositional equivalence is an equivalence relation on the set of all compound propositions.
4. Give a description of each of the congruence classes modulo 6.
5. Which of the following collections of subsets are partitions of {1, 2, 3, 4, 5, 6} ?
6. Consider the equivalence relation on the set of integers R = { (x, y) | x - y is an integer }.