Page 201 - HOW TO PROVE IT: A Structured Approach, Second Edition

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More About Relations 187
5. The following diagram shows two relations R and S. Find S ◦ R.
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6. Suppose r and s are two positive real numbers. Let D r and D s be defined
as in part 3 of Example 4.3.1. What is D r ◦ D s ? Justify your answer with
a proof. (Hint: In your proof, you may find it helpful to use the triangle
inequality; see exercise 12(c) of Section 3.5.)
7. Prove part 1 of Theorem 4.3.4.
∗
8. Prove part 3 of Theorem 4.3.4.
9. Suppose A and B are two sets.
(a) Show that for every relation R from A to B, R ◦ i A = R.
(b) Show that for every relation R from A to B, i B ◦ R = R.
10. Suppose S is a relation on A. Let D = Dom(S) and R = Ran(S). Prove
∗
−1
that i D ⊆ S −1 ◦ S and i R ⊆ S ◦ S .
11. Suppose R is a relation on A. Prove that if R is reflexive then R ⊆ R ◦ R.
12. Suppose R is a relation on A.
−1
(a) Prove that if R is reflexive, then so is R .
−1
(b) Prove that if R is symmetric, then so is R .
−1
(c) Prove that if R is transitive, then so is R .
13. 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?
14. 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?