Problem 1. Define the relation R on the set of all integers by i R j <=> (i-j) is even. Prove or disprove: R is an equivalence relation.
Problem 2.
Define the relation R on the set of all integers by
i R j <=> i is a divisor of j. Which of the properties
(reflexive, irreflexive, symmetric, asymmetric, antisymmetric, transitive)
does this relation possess? Prove each answer (i.e., show why the property
holds or give a counterexample demonstrating that it does not hold).
Problem 3.
A graph is said to be complete if there is an edge in it between
any two vertices; the complete graph on n vertices is denoted by
Kn. A graph is bipartite if we can partition its
set of vertices V into two subsets V1 and
V2, such that the only edges of the graph are between
vertices of V1 and vertices of V2 --
without any edge among the vertices of V1 and without
any edge among the vertices of V2. A bipartite graph
is complete if there is an edge between any vertex of V1
and any vertex of V2;
if V1 has m vertices and V2
has n vertices, we denote the complete bipartite graph on these
vertices by Km,n.
Problem 4.
Give an example of an equivalence relation on the set of all positive integers
that has an infinite number of equivalence classes, with each equivalence class
of infinite size. (Hint: it is probably easiest to devise such a class
by considering first the classes, then defining the relation from the
classes, rather than vice versa; you may also want to define a collection of
classes that may not form a partition because they leave some numbers out,
then put all remaining number into one class; finally, think of some feature
of numbers that you know about and that is shared among an infinity of numbers,
then use this feature to define classes and proceed as recommended.)
Problem 5.
The adjacency matrix below describes a directed graph.
1 2 3 4 5 6 7 8 9 10
_______________________
1 | 0 1 0 0 0 0 0 0 0 1 |
2 | 0 0 1 0 1 0 0 0 0 0 |
3 | 1 0 0 0 0 0 0 0 0 0 |
4 | 0 0 0 0 0 1 0 0 0 0 |
5 | 0 0 0 0 0 0 0 0 0 1 |
6 | 0 0 0 0 0 0 0 1 0 0 |
7 | 0 0 0 0 0 0 0 0 1 0 |
8 | 0 0 0 1 0 0 0 0 0 0 |
9 | 0 0 0 0 0 0 1 0 0 0 |
10 | 0 0 1 0 0 0 0 0 0 0 |
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