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Homework Nine Part II (of II)
Show that Q+ is countable and use this fact to show that Q is
countable.
Hint. This is a generalization of the proof that N × N is
countable. If you are truly stuck, take a look at pages 247 − 250
in the textbook.
Let A be an uncountable set and let B ⊂ A be a countable
subset of A. Prove that |A| = |A − B|.
Hint. This is a generalization of one of the last problems
from Workshop 5 with a very similar solution.
(a) Let Pfin (N) denote the set of finite subsets of N, show that
Pfin (N) is countable.
(b) Let Pinf (N) denote the set of infinite subsets of N. Show
that Pinf (N) is uncountable.
Hint. Use the previous problem.
A cycle in a graph is a path whose initial and terminal vertices
coincide. A graph with no cycles is called a forest. Show that
if a forest has v vertices, e edges and c connected components,
then v = e + c. Think about why this is surprising and weird.
A graph is called bipartite if its set of vertices can be separated
into two disjoint sets U and V such that every edge in the
graph connects a vertex in U to a vertex in V . Prove that a
finite graph is bipartite if and only if it contains no cycles of
odd length.
A colouring of a graph G is a function θ : V (G) → C from
V (G) to a finite set of colours C := {c1 , c2 , c3 , . . . , ck } such that
no two adjacent vertices are assigned the same colour. If G can
be coloured with a set of k colours, then we say G can be kcoloured. The minimal value of k for which G can be k-coloured
is the chromatic number of G and is denoted by χ(G).
(a) Find χ(G) if G is a cycle with at least three vertices.
(b) Find χ(G) if G is a complete graph.
(c) Find χ(G) if G is a tree with at least one edge.
(d) Find χ(G) if G is bipartite with at least one edge.
(e) Find χ(G) for your favourite graph G.
(f) Describe a “real world” problem that can be modelled by
a graph G where knowing χ(G) is useful.
Hint. As usual, it may be useful to draw several small examples
before describing the general cases. Your answers may be given
as a function of |V (G)| or as a function of the parity of |V (G)|.
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