Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 424 — #432
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Chapter 10
Directed graphs & Partial Orders
Homework Problems
Problem 10.58.
Let R 1 and R 2 be two equivalence relations on a set A. Prove or give a counterexample
to the claims that the following are also equivalence relations:
(a) R 1 \ R 2 .
(b) R 1 [ R 2 .
Problem 10.59.
Prove that for any nonempty set D, there is a unique binary relation on D that is
both a weak partial order and also an equivalence relation.
Exam Problems
Problem 10.60.
Let A be a nonempty set.
(a) Describe a single relation on A that is both an equivalence relation and a weak
partial order on A.
(b) Prove that the relation of part (a) is the only relation on A with these properties.