Page 237 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Equivalence Relations 223
4. Which of the following relations on R are equivalence relations? For those
that are equivalence relations, what are the equivalence classes?
(a) R ={(x, y) ∈ R × R | x − y ∈ N}.
(b) S ={(x, y) ∈ R × R | x − y ∈ Q}.
n
(c) T ={(x, y) ∈ R × R |∃n ∈ Z(y = x · 10 )}.
5. In the discussion of the same-birthday equivalence relation B, we claimed
∗
that P/B ={P d | d ∈ D}. Give a careful proof of this claim. You will find
when you work out the proof that there is an assumption you must make
about people’s birthdays (a very reasonable assumption) to make the proof
work. What is this assumption?
6. Let T be the set of all triangles, and let S ={(s, t) ∈ T × T | the triangles
s and t are similar}. (Recall that two triangles are similar if the angles of
one triangle are equal to the corresponding angles of the other.) Verify
that S is an equivalence relation.
7. Complete the proof of Lemma 4.6.7.
8. Suppose R and S are equivalence relations on A and A/R = A/S. Prove
that R = S.
9. Suppose R is an equivalence relation on A. Let F = A/R, and let S be the
∗
equivalence relation determined by F. In other words, S =∪ X∈F (X ×
X). Prove that S = R.
10. Let C m be the congruence mod m relation defined in the text, for a positive
integer m.
(a) CompletetheproofthatC m isanequivalencerelationonZbyshowing
that it is reflexive and symmetric.
(b) Find all the equivalence classes for C 2 and C 3 . How many equivalence
classes are there in each case? In general, how many equivalence
classes do you think there are for C m ?
2
2
11. Prove that for every integer n, either n ≡ 0 (mod 4) or n ≡ 1 (mod 4).
∗ 12. Suppose m is a positive integer. Prove that for all integers a, b, c, and d,
if a ≡ c (mod m) and b ≡ d (mod m) then a + b ≡ c + d (mod m) and
ab ≡ cd (mod m).
13. Suppose R is an equivalence relation on A and B ⊆ A. Let S = R ∩
(B × B).
(a) Prove that S is an equivalence relation on B.
(b) Prove that for all x ∈ B,[x] S = [x] R ∩ B.
14. Suppose B ⊆ A, and define a relation R on P (A) as follows:

R ={(X, Y) ∈ P (A) × P (A) | (X Y) ⊆ B}.
(a) Prove that R is an equivalence relation on P (A).