Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 419 — #427
10.11. Summary of Relational Properties 419
corresponding to the empty set must be scheduled first because ; S for every
nonempty set S f1; 2; 3; 4; 5g.
(a) What is the minimum parallel time to complete these tasks?
(b) Describe a maximum size antichain in this partial order.
(c) Briefly explain why the minimum number of processors required to complete
these tasks in minimum parallel time is equal to the size of the maximum antichain.
Problem 10.48.
Let R be a weak partial order on a set A. Suppose C is a finite chain. 17
(a) Prove that C has a maximum element. Hint: Induction on the size of C .
(b) Conclude that there is a unique sequence of all the elements of C that is strictly
increasing.
Hint: Induction on the size of C , using part (a).
Problems for Section 10.9
Practice Problems
Problem 10.49.
Verify that if either of R 1 or R 2 is irreflexive, then so is R 1 R 2 .
Class Problems
Problem 10.50.
Let R 1 , R 2 be binary relations on the same set A. A relational property is preserved
under product, if R 1 R 2 has the property whenever both R 1 and R 2 have the
property.
(a) Verify that each of the following properties are preserved under product.
1. reflexivity,
2. antisymmetry,
3. transitivity.
17 A set C is a chain when it is nonempty, and all elements c; d 2 C are comparable. Elements c
and d are comparable iff Œc R d OR d R c.