The equivalence relation on Z given by (?, ?) ? ? iff ? ? ? mod? is an
equivalence relation for an integer ? ? 2.
a) What are the equivalence classes for R given a fixed integer ? ?2?
b) We denote the set of equivalence classes you found in (a) byZ_5. Even though elements of Z_5 are
sets, it turns out that we can define addition and multiplicationin the expected ways: [?] + [?] = [? + ?] and [?] ? [?] =[??]
Construct the addition and multiplication tables for Z_4 and Z_5.Record sums and products in the
form [r], where 0 ? r ? 3 (or 4, respectively).
c) Let [?], [?] ? Z_10. If [?][?] = [0], does it follow that [?] =[0] or [?] = [0]?
d) How would you answer the question from (c) for Z_11, Z_12?,Z_13?
e) For which integers ? ? 2 is the following statement true?
“Let [?], [?] ? Z_5. If [?][?] = [0], then [?] = [0] or [?] =[0].”
