For a real number x, define r(x) to be the closest integer to x. More precisely, if x ∈ (n - 0.5, n + 0.5]for some n ∈ Z, then r(x) = n. Define the relation ∼ on R by x ∼ y if r(x) = r(y).
a) Show that r : R → Z is well defined. In other words, show that if x ∈ R, and m, n ∈ Z satisfy r(x) = m and r(x) = n, then n = m.
b) Show that ∼ is an equivalence relation on R.
c) Give the set [2.718].