70
Solutions
to
Exercises
20.
(a)
[BB]
This set is uncountable. The function defined
by
f(x)
=
xI
gives a onetoone corre
spondence between it and
(0,1),
which we showed in the text
to
be
uncountable.
(b) This set is countably infinite. Just follow the sequence given in the text for the set
of
all positive
rationals, but omit any rational number not in
(1,2).
(c) This set is finite.
In
fact,
it
contains at most
99
2
elements since there are
99
possible numerators
and, for each numerator,
99
possible denominators.
(d) This set is countably infinite. List the elements
as
follows (deleting any repetitions such as
5/100 
1/20)·
99 99
99
98 98
98

·6'7'···'
104'6'7'···'W4'···
.
(e)
[BB]
This set is countably infinite.
In
Exercise 3(c)
we
showed
it
is in onetoone correspondence
with
N
x
N.
(f)
This set is countably infinite because it is in onetoone correspondence with
Q
via the function
f
defined by
f(a,
b)
=
a.
(g) This set is uncountable. It is in onetoone correspondence with
[1,1]
=
{x
I
1
~
x
~
I}
via
the function
f
defined by
f(a,
b)
=
a.
The interval
[1,1]
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 Summer '10
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 Set Theory, Graph Theory, Natural number, onetoone correspondence, Countable set

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