Define the relation

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For a real number x, define f(x) to be the fractional part of x. More precisely, if x ∈ [n, n + 1) for some n ∈ Z, then f(x) = x - n. Define the relation ∼ on R by x ∼ y if f(x) = f(y).

a) Show that m : R → [0, 1) is well defined. In other words, show that if x ∈ R, and a, b ∈ Z satisfy f(x) = a and f(x) = b, then a = b.

b) Show that ∼ is an equivalence relation on R.

c) Give the set [2.718]

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