Question

2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for all X, Y E P (s), (X, Y) E R if and only if IX-T] = IY-T]. (a) Prove that R is an equivalence relation. (b) How many equivalence classes are there? Explain. (c) How mauy elements of [ø], the equivalence class of ø, are there? Explain (d) How many elements of [f1,2,3, 4)], the equivalence class of (1,2,3, 4), are there? Explain. 3. Let A (1,2,3,4). For any function f : A A and any relation R on A, we define the relation S on A by: for any a, beA, aSb if and only if (f (a), f (b)) R. For each of the following statements, prove or disprove the statement. (a) For all functions f A A and all relations R on A, if S is reflexive then R is reflexive. (b) For all functions f : A A and all relations R on A, if R is reflexive then S is reflexive. c) For all functionsf: A A and all relations R on A, if S is symmetric then R is symmetric. (d) For all finctions :A A and all relations R on A, if R is symmetric then S is symmetric.

Answer 1

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