CS173: Discrete Mathematical Structures
Spring 2006
Homework #12
Due 04/30/06, 8am
Relation Basics
1. Determine the reflexivity, symmetry, and transitivity of the following relations:
a. R=โ
on set {0,1,2,4,6}S=.
b. {(0,1),(1,0),(2,4),(4,2),(4,6),(6,4)}R=on set {0,1,2,4,6}S
=
i. Determine its reflexivity, symmetry, and transitivity
ii. What is its transitive closure?
c.
x
Ry x y
โ
โis a multiple of 3 on setS
=
๎
d. ||||
x
R
y
x
y
โ
โ on set {1,2,3,4,5,6,7,8,9}S
=
e. 11 2 2 1 21 2
(, )( , ) ,
xy
Rx
y
xx
yy
โโค โฅon set S
=
ร๎ ๎
2. 1
R
and 2
R
are two relations on a set S.
a. If 1
R
and 2
R
are reflexive, is 12
R
Rโฉreflexive? Is 12
R
Rโชreflexive? Explain
your answer.
b. If 1
R
and 2
R
are transitive, is 12
R
Rโฉtransitive? Is 12
R
Rโชtransitive? Explain
your answer.
c. Can 1
R
be symmetric and antisymmetric at the same time? Why or why
not?
d. Can 1
Rbe neither symmetric nor antisymmetric? Why or why not?
Equivalence Relations
3.
a. Prove that a relation R defined as (,) (, )abRcd a d b c
โ
+=+is an
equivalence relation on
ร
๎๎
.
b. Determine the number of equivalence classes for R under set
{0, 1, 2 , , } {0, 1, 2, , }nnโ
โ
โ
ร โ
โ
โ
.
Docsity.com
