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Question: betore attempting these problems make sure to read the handont...

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Betore attempting these problems, make sure to read the handont on relations, which was uploaded to Canvas 1. Let S denote the Cartesian product of Z and Z(o, i.e., S= {(n, d) : n € Ζ.de Z\(0)). Let ~ be the relation on S given by (ni,di) (r2, d2) if and only if nd2 di (a) Prove that~ is an equivalence relation on S. (b) Let Eind) denote the equivalence class containing the element (n,d) e s, and let Q denote the set of all such equivalence classes, i.e., Let e and o be binary operations on Q given by Give a rigorous mathematical formulation of what it means to say that that 9 and are well-defined. and G are well-defined, and then prove (c) Let f :Z +Q be defined by f(n) E(m,1) Prove that f is an injection, and that This allows us to think of the set as a copy of 2 that is contained in Q. and the g the uotati on 2 to denote the equivalence c e the statements in parts (b) and () using the notation 3 to denote the equivalence class End), . notations + and . in place of φ and o, respectively, we henceforth adopt this notation when dealing with

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