Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 413 — #421
10.11. Summary of Relational Properties 413
Problems for Section 10.6
Exam Problems
Problem 10.31.
Prove that for any nonempty set D, there is a unique binary relation on D that is
both asymmetric and symmetric.
Problem 10.32.
Let D be a set of size n > 0. Shown that there are exactly 2 n binary relations on
D that are both symmetric and antisymmetric.
Homework Problems
Problem 10.33.
Prove that if R is a transitive binary relation on a set A then R D R C .
Class Problems
Problem 10.34.
Let R be a binary relation on a set D. Each of the following equalities and containments
expresses the fact that R has one of the basic relational properties: reflexive,
irreflexive, symmetric, asymmetric, antisymmetric, transitive. Identify which property
is expressed by each of these formulas and explain your reasoning.
(a) R \ Id D D ;
(b) R R 1
(c) R D R 1
(d) Id D R
(e) R ı R R
(f) R \ R 1 D ;
(g) R \ R 1 Id D