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2. Fix m, n ? ?. Define a mapping f : ?/n? ? ?/m? by f([a]_n) = [a]_m. a. Prove that if m | n then f is a well-defined function. That is, prove that if [a]_n = [b]_n then f([a]_n) = f([b]_n). b. Let n = 12 and m = 3. Write PreIm_f({[1]_3, [2]_3}) in roster notation. c. Suppose m ? n. Show that f is ill-defined. That is, show there exist a, b ? ? such that [a]_n = [b]_n but f([a]_n) ? f([b]_n).

          2. Fix m, n ? ?. Define a mapping f : ?/n? ? ?/m? by f([a]_n) = [a]_m.
a. Prove that if m | n then f is a well-defined function. That is, prove that if [a]_n = [b]_n then f([a]_n) = f([b]_n).
b. Let n = 12 and m = 3. Write PreIm_f({[1]_3, [2]_3}) in roster notation.
c. Suppose m ? n. Show that f is ill-defined. That is, show there exist a, b ? ? such that [a]_n = [b]_n but f([a]_n) ? f([b]_n).

2. Fix m, n ? ?. Define a mapping f : ?/n? ? ?/m? by f([a]n) = [a]m.
a. Prove that if m | n then f is a well-defined function. That is, prove that if [a]n = [b]n then f([a]n) = f([b]n).
b. Let n = 12 and m = 3. Write PreImf([1]3, [2]3) in roster notation.
c. Suppose m ? n. Show that f is ill-defined. That is, show there exist a, b ? ? such that [a]n = [b]n but f([a]n) ? f([b]n).

Added by Richard S.

Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert Sri K

04/30/2022

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Fix m, n ∈ ℕ. Define a mapping f : ℤ/nℤ → ℤ/mℤ by f([a]_n) = [a]_m. a. Prove that if m | n then f is a well-defined function. That is, prove that if [a]_n = [b]_n then f([a]_n) = f([b]_n). b. Let n = 12 and m = 3. Write PreIm_f({[1]_3, [2]_3}) in roster notation. c. Suppose m ∤ n. Show that f is ill-defined. That is, show there exist a, b ∈ ℤ such that [a]_n = [b]_n but f([a]_n) ≠ f([b]_n).