Rob Bayer Math 55 Worksheet August 3, 2009
Relations
1. Determine whether each of the following relations are symmetric, reflexive, transitive, and/or antisymmetric.
(a) <on N
(b) ā¤on N
(c) The relation on Rdefined by aā¼biff aābāZ
(d) The relation on people given by āis a child ofā
(e) x=y2on R
(f) 6=
2. Whatās the probability that a randomly chosen relation on a set of size nis
(a) Symmetric? (b) Reflexive? (c) Both?
3. A relation is called irreflexive if (a, a)/āRfor any a
(a) Give a natural example of an irreflexive relation
(b) Give a (semi-)natural example of a relation that is neither reflexive nor irreflexive
4. Let R1be the relation ācongruent mod 3ā and R2be the relation ācongruent mod 4,ā both on Z. What are the
relations
(a) R1āŖR2
(b) R1ā©R2
(c) R2āR1
5. True/False. For those that are true, prove it. For those that are false, provide a counterexample.
(a) The intersection of two symmetric relations is symmetric
(b) The union of two antisymmetric relations is antisymmetric
(c) The intersection of two transitive relations is transitive
(d) The union of two transitive relations is transitive
6. (a) Whatās wrong with the following āproofā that transitive + symmetric āreflexive.
āLet ā¼be a transitive symmetric relation. Then we know aā¼bābā¼aso by transitivity, aā¼aand thus
ā¼must be reflexiveā
(b) Give an example of a relation that is transitive and symmetric but not reflexive, thus showing that the
āresultā from part (a) in fact false.
Representing Relations
1. What relation is represented by each of the following matrices? Assume the set is {1,2,3}and the columns and
rows are listed in increasing order
(a) ļ£®
ļ£°
101
010
101
ļ£¹
ļ£»
(b) ļ£®
ļ£°
111
010
110
ļ£¹
ļ£»
(c) ļ£®
ļ£°
010
001
100
ļ£¹
ļ£»
2. Determine whether each of the relations from problem 1 are transitive, symmetric, reflexive, irreflexive and/or
antisymmetric.
3. If Ris a relation, then the relation Rā1is defined by aRā1bābRa. How does the matrix for Rrelate to that
of Rā1? What about for R, the complement of R?