Page 238 - HOW TO PROVE IT: A Structured Approach, Second Edition

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224 Relations
(b) Prove that for every X ∈ P (A) there is exactly one Y ∈ [X] R such
that Y ∩ B = ∅.
∗ 15. Suppose F is a partition of A, G is a partition of B, and A and B are
disjoint. Prove that F ∪ G is a partition of A ∪ B.
16. Suppose R is an equivalence relation on A, S is an equivalence relation
on B, and A and B are disjoint.
(a) Prove that R ∪ S is an equivalence relation on A ∪ B.
(b) Prove that for all x ∈ A,[x] R∪S = [x] R , and for all y ∈ B,[y] R∪S =
[y] S .
(c) Prove that (A ∪ B)/(R ∪ S) = (A/R) ∪ (B/S).
17. Suppose F and G are partitions of a set A. We define a new family of sets
F · G as follows:
F · G ={Z ∈ P (A) | Z = ∅ and ∃X ∈ F∃Y ∈ G(Z = X ∩ Y)}.
Prove that F · G is a partition of A.
− +
18 Let F ={R , R , {0}} and G ={Z, R \ Z}, and note that both F and G
are partitions of R. List the elements of F · G. (See exercise 17 for the
meaning of the notation used here.)
∗
19. Suppose R and S are equivalence relations on a set A. Let T = R ∩ S.
(a) Prove that T is an equivalence relation on A.
(b) Prove that for all x ∈ A,[x] T = [x] R ∩ [x] S .
(c) Prove that A/T = (A/R) · (A/S). (See exercise 17 for the meaning
of the notation used here.)
20. Suppose F is a partition of A and G is a partition of B. We define a new
family of sets F ⊗ G as follows:
F ⊗ G ={Z ∈ P (A × B) |∃X ∈ F∃Y ∈ G(Z = X × Y)}.
Prove that F ⊗ G is a partition of A × B.
21. Let F ={R , R , {0}}, which is a partition of R. List the elements of
∗ − +
F ⊗ F, and describe them geometrically as subsets of the xy-plane. (See
exercise 20 for the meaning of the notation used here.)
22. Suppose R is an equivalence relation on A and S is an equivalence relation
on B. Define a relation T on A × B as follows:

T ={((a, b), (a , b )) ∈ (A × B) × (A × B) | aRa and bSb }.
(a) Prove that T is an equivalence relation on A × B.
(b) Prove that if a ∈ A and b ∈ B then [(a, b)] T = [a] R × [b] S .
(c) Prove that (A × B)/T = (A/R) ⊗ (B/S). (See exercise 20 for the
meaning of the notation used here.)