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building a sequence which includes each of the
positive rationals
exactly once. What are the first four terms of this sequence?
If, further, the rational number
· 5,
show how to construct a sequence which runs through all the
rational numbers once. What are the first six terms of this
sequence?
a
/
b
is matched with 2
· 3
26 (Optional) If
a
,
b
,
c
and
d
are positive integers matching the
number (
a
/
b
)
(
c
/
d
)
2 with thenatural numbr 2
· 3
· 5
· 7
gives
a one-to-one matching of the numbers
x
y
2 (where
x
and
y
are
positiverationals) with a subst of N, provided the fractions
a
/
b
and
c
/
d
are in their lowest terms. Use this matching to construct a
sequence containing all the numbers of the form
x
y
2, where
x
and
y
arepositiverationals. List thefirst six numbrs in your
sequence.
When it is possible to put the elements of a set into a sequence without
repetition, the set is said to be
countably infinite
. There is then a
one-to-one correspondence between the elements of the set and the
natural numbers N.
27 Which of thefollowing sts arecountably infinit: Z,
T
(of qn 8),
D
(of qn 9), Q, the set of infinite decimals between 0 and 1?
When an infinite set is not
countably infinite
it is said to be
uncountable
.
We will shortly claim as a principle that every infinite decimal is a
number, that is that every infinite decimal sequence is convergent. The
set of infinite decimals is then known as the set of real numbers and
denoted by R.
The contrast between the
countably infinite
rational numbers and
the
uncountable
real numbers is remarkable. A countably infinite set of
numbers can be shown to occupy only a small portion of the real line.
28 If for each
n
N, theintrval
x
n
1/2
x
n
1/2
is deleted
from the number line, is every element of
N
deleted? What is the
total length deleted?
29 Let
be a fixed positive number. If for each
n
N, theintrval
x
n
/2
x
n
/2
is deleted from the number line, is
every element of N deleted? What is the total length deleted? Can
you make this length as small as you like? How much length
should wesay thest
N
occupies?
30 Let (
a
) be any sequence of numbers and let
beany positive
number. If for each
n
N, theintrval
/2
is deleted from the number line, is every element of the sequence
(
a
x
a
/2
x
a
) deleted? What is the total length deleted? Deduce that a
