The answers to exercises marked IBB] can be found in the Back of the Book.
Define ~ on R2 by (a, y) ~ (u, v) if and only if x − y = u − v.
(a) [BB| Criticize and then correct the following “proof” that ~ is reilexive.
“If (x. y) ~ (a, v), then x − y = x − v, which is true."
(b) What is wrong with the following interpretation of symmetry in this situation?
“If (a, y) ∈ R, then (y, x) ∈ R”
Write a correct statement of the symmetric property (as it applies to the relation ~ in this exercise).
(c) Criticize and then correct the following “proof” that ~ is symmetric.
“(x, y) ~ (u, v) if x - y = u - v. Then u − v =x − y. So (zr, u) ~ (x, y).”
(d) Criticize and correct the following “proof” of transitivity.
“(x, y) ~ (u, v) and (u, v) ~ {w, z). Then ti − v = w − z, so if r − y = u − v, then x − y = w − z. So (x, y) ~ (w, z).”
(e) Why does ~ define an equivalence relation on R2?
(f) Determine the equivalence classes of (0,0) and (2, 3) and describe these geometrically.
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