3. Let a relation be given on A = Z (Z - {0}) by (n, m)~...

   

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3. Let a relation be given on A = Z× (Z - {0}) by (n, m)~ (k, 1) if and only if nl = mk. (a) Prove this is an equivalence relation. (b) Describe the equivalence class of (2,3). (c) Define an addition operation on the elements of A by (n, m) + (k, 1) = (nl + mk, ml). Show that if (n, m) ~ (n', m') and (k, l) ~ (k', l'), then (n, m) + (k, l) ~ (n', m') + (k', l'). We call this " respects addition." 3. Let a relation be given on A = Z× (Z - {0}) by (n, m)~ (k, 1) if and only if nl = mk. (a) Prove this is an equivalence relation. (b) Describe the equivalence class of (2,3). (c) Define an addition operation on the elements of A by (n, m) + (k, 1) = (nl + mk, ml). Show that if (n, m) ~ (n', m') and (k, l) ~ (k', l'), then (n, m) + (k, l) ~ (n', m') + (k', l'). We call this " respects addition."