(1)Let1and2be two equivalence relations on the same setA. Assume that there are at least two equivalence classes for1, and that any two elements that are in different equivalence classes of1are in the same equivalence class of2. As completely as possible, describe the equivalence classes of2, and prove your assertion.

(2)The following argument attempts to show that a relationonAthat is both symmetric and transitive is also reflexive. Given aA, from ab and symmetry we get ba. From both ab and ba, transitivity givesaa. Thus,aafor allaA, sois reflexive.

• Findanexamplethatshowsthattheassertionisfalse;thatis,findarelationthatissymmetricandtransitive
• but not reflexive. (You can give the example as a set of ordered pairs.)
• Identify precisely the flaw in the argument. (The flaw is not that the conclusion is false; instead, it is some
• statement or implicit assumption in the attempted proof that is not justifiable. You need to identify that
• unjustifiable statement. Examine every small step in the proof and ask yourself whether it is valid.)

(3)IsthefollowingstatementtrueforallsetsA,B,C,andD?IfABCD,thenACandBD. Provide a proof if you claim that this statement is correct; provide a counterexample if you claim that this statement is false.

(4)On Homework#8, you showed that if f:AB is injective, then f(YX)=f(Y)f(X) for all sets X and Ywith XYA.

(a)State the converse of this result.

(b)Determine whether the converse is true. If you claim that it is true, then prove it. If you claim that it is false, then give a counterexample showing a case where it fails.