Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 416 — #424
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Chapter 10
Directed graphs & Partial Orders
(a) The superset relation, on the power set pow f1; 2; 3; 4; 5g.
(b) The relation between any two nonnegative integers a, b given by a b
.mod 8/.
(c) The relation between propositional formulas G, H given by G IMPLIES H is
valid.
(d) The relation ’beats’ on Rock, Paper and Scissor (for those who don’t know the
game “Rock, Paper, Scissors:” Rock beats Scissors, Scissors beats Paper and Paper
beats Rock).
(e) The empty relation on the set of real numbers.
(f) The identity relation on the set of integers.
Problem 10.39. (a) Verify that the divisibility relation on the set of nonnegative
integers is a weak partial order.
(b) What about the divisibility relation on the set of integers?
Problem 10.40.
Prove directly from the definitions (without appealing to DAG properties) that if a
binary relation R on a set A is transitive and irreflexive, then it is asymmetric.
Class Problems
Problem 10.41.
Show that the set of nonnegative integers partially ordered under the divides relation.
. .
(a) . . . has a minimum element.
(b) . . . has a maximum element.
(c) . . . has an infinite chain.
(d) . . . has an infinite antichain.
(e) What are the minimal elements of divisibility on the integers greater than 1?
What are the maximal elements?