(Solved): Solve #10 according to the hint below: 9. Suppose that A is a nonempty set, and f is a function t ...

Solve #10 according to the hint below:

9. 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) such that f(x) = f(y). a) Show that R is an equivalence relation on A. b) What are the equivalence classes of R? 10. Suppose that A is a nonempty set and R is an equivalence relation on A. Show that there is a function f with A as its domain such that (x, y) E R if and only if f(x) = f(y). = Although the situation in Exer 9 is not exactly the same as here, be sure to do that exercise first to warm-up. (In Exer 9 the function f is given 1st, and then a relation R is to be defined in terms of f. Here the relation R is given 1st, and then a function f is to be defined in terms of R.) Here no codomain B is being given, but to solve the problem you will need to specify a particular set B. Outline for creating an f that works: Choose the codomain B for f to be P(A), the power set of A. Then you need to specify a particular rule to define a function f that assigns a particular subset of A to each element of A. (You'll need to refer to R, of course.) To prove that your f has the required property, consider Theorem 9.5.1. THEOREM 1 Let R be an equivalence relation on a set A. These statements for elements a and b of A are equivalent: (i) arb (ii) [a] = [b] (iii) [a] n [b] 0