Page 202 - HOW TO PROVE IT: A Structured Approach, Second Edition

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188 Relations
15. Suppose R 1 and R 2 are relations on A. For each part, give either a proof
or a counterexample to justify your answer.
(a) If R 1 and R 2 are reflexive, must R 1 \ R 2 be reflexive?
(b) If R 1 and R 2 are symmetric, must R 1 \ R 2 be symmetric?
(c) If R 1 and R 2 are transitive, must R 1 \ R 2 be transitive?
16. Suppose R and S are reflexive relations on A. Prove that R ◦ S is reflexive.
∗ 17. Suppose R and S are symmetric relations on A. Prove that R ◦ S is sym-
metric iff R ◦ S = S ◦ R.
18. Suppose R and S are transitive relations on A. Prove that if S ◦ R ⊆ R ◦ S
then R ◦ S is transitive.
19. Consider the following putative theorem.
Theorem? Suppose R is a relation on A, and define a relation S on P (A)
as follows:
S ={(X, Y) ∈ P (A) × P (A) |∃x ∈ X∃y ∈ Y(xRy)}.

If R is transitive, then so is S.
(a) What’s wrong with the following proof of the theorem?
Proof. Suppose R is transitive. Suppose (X, Y) ∈ S and (Y, Z) ∈ S.
Then by the definition of S, xRy and yRz, where x ∈ X, y ∈ Y,
and z ∈ Z. Since xRy, yRz, and R is transitive, xRz. But then since
x ∈ X and z ∈ Z, it follows from the definition of S that (X, Z) ∈ S.
Thus, S is transitive.
(b) Is the theorem correct? Justify your answer with either a proof or a
counterexample.
∗
20. Suppose R is a relation on A. Let B ={X ∈ P (A) | X = ∅}, and define
a relation S on B as follows:
S ={(X, Y) ∈ B × B |∀x ∈ X∀y ∈ Y(xRy)}.
Prove that if R is transitive, then so is S. Why did the empty set have to
be excluded from the set B to make this proof work?
21. Suppose R is a relation on A, and define a relation S on P (A) as follows:

S ={(X, Y) ∈ P (A) × P (A) |∀x ∈ X∃y ∈ Y(xRy)}.
For each part, give either a proof or a counterexample to justify your
answer.
(a) If R is reflexive, must S be reflexive?
(b) If R is symmetric, must S be symmetric?
(c) if R is transitive, must S be transitive?