Page 239 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Equivalence Relations 225
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23. Suppose R and S are relations on a set A, and S is an equivalence relation.

We will say that R is compatible with S if for all x, y, x , and y in A,if

xSx and ySy then xRy iff x Ry .

(a) Prove that if R is compatible with S, then there is a unique relation
T on A/S such that for all x and y in A,[x] S T [y] S iff xRy.
(b) Suppose T is a relation on A/S and for all x and y in A,[x] S T [y] S
iff xRy. Prove that R is compatible with S.
24. Suppose R is a relation on A and R is reflexive and transitive. (Such a
−1
relation is called a preorder on A.) Let S = R ∩ R .
(a) Prove that S is an equivalence relation on A.
(b) Prove that there is a unique relation T on A/S such that for all x and
y in A,[x] S T [y] S iff xRy. (Hint: Use exercise 23.)
(c) Prove that T is a partial order on A/S, where T is the relation from
part (b).
25. Let I ={1, 2,..., 100}, A = P (I), and R ={(X, Y) ∈ A × A | Y has
at least as many elements as X}.
(a) Prove that R is a preorder on A. (See exercise 24 for the definition of
preorder.)
(b) Let S and T be defined as in exercise 24. Describe the elements of
A/S and the partial order T . How many elements does A/S have? Is
T a total order?
26. Suppose A is a set. If F and G are partitions of A, then we’ll say that F
refines G if ∀X ∈ F∃Y ∈ G(X ⊆ Y). Let P be the set of all partitions of
A, and let R ={(F, G) ∈ P × P | F refines G}.
(a) Prove that R is a partial order on P.
(b) Suppose that S and T are equivalence relations on A. Let F = A/S
and G = A/T . Prove that S ⊆ T iff F refines G.
(c) Suppose F and G are partitions of A. Prove that F · G is the greatest
lower bound of the set {F, G} in the partial order R. (See exercise 17
for the meaning of the notation used here.)