The equivalence relation on Z given by (?, ?) ∈ ? iff ? ≡ ? mod...

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) by Z_5. Even though elements of Z_5 are
sets, it turns out that we can define addition and multiplication in 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].”

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Expert Solution

Colby Messinger answered 1 year ago

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