CS 702
Discrete Mathematics and Probability Theory
Spring 2009
Alistair Sinclair, David Tse
Note 21
Infinity and Countability
Consider a function (or mapping)
f
that maps elements of a set
A
(called the
domain
of
f
) to elements of set
B
(called the
range
of
f
). For each element
x
∈
A
(“input”),
f
must specify one element
f
(
x
)
∈
B
(“output”).
Recall that we write this as
f
:
A
→
B
. We say that
f
is a
bijection
if every element
a
∈
A
has a unique
image
b
=
f
(
a
)
∈
B
, and every element
b
∈
B
has a unique
preimage a
∈
A
:
f
(
a
) =
b
.
f
is a
onetoone function
(or an
injection
) if
f
maps distinct inputs to distinct outputs. More rigorously,
f
is onetoone if the following holds:
x
negationslash
=
y
⇒
f
(
x
)
negationslash
=
f
(
y
)
.
The next property we are interested in is functions that are
onto
(or
surjective
).
A function that is onto
essentially “hits” every element in the range (i.e., each element in the range has at least one preimage).
More precisely, a function
f
is onto if the following holds:
∀
y
∃
x
:
f
(
x
) =
y
. Here are some examples to
help visualize onetoone and onto functions:
Onetoone
Onto
Note that according to our definition a function is a bijection iff it is both onetoone and onto.
Cardinality
How can we determine whether two sets have the same
cardinality
(or “size”)? The answer to this question,
reassuringly, lies in early grade school memories: by demonstrating a
pairing
between elements of the two
sets. More formally, we need to demonstrate a
bijection f
between the two sets. The bijection sets up a
onetoone correspondence, or pairing, between elements of the two sets. We know how this works for finite
sets. In this lecture, we will see what it tells us about
infinite
sets.
Are there more natural numbers
N
than there are positive integers
Z
+
? It is tempting to answer yes, since
every positive integer is also a natural number, but the natural numbers have one extra element 0
/
∈
Z
+
. Upon
more careful observation, though, we see that we can generate a mapping between the natural numbers and
the positive integers as follows:
N
0
1
2
3
4
5
. . .
↓
arrowsoutheast
arrowsoutheast
arrowsoutheast
arrowsoutheast
arrowsoutheast
arrowsoutheast
Z
+
1
2
3
4
5
6
. . .
CS 702, Spring 2009, Note 21
1
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Why is this mapping a bijection? Clearly, the function
f
:
N
→
Z
+
is onto because every positive integer
is hit. And it is also onetoone because no two natural numbers have the same image. (The image of
n
is
f
(
n
) =
n
+
1, so if
f
(
n
) =
f
(
m
)
then we must have
n
=
m
.) Since we have shown a bijection between
N
and
Z
+
, this tells us that there are as many natural numbers as there are positive integers! Informally, we have
proved that “
∞
+
1
=
∞
.”
What about the set of
even
natural numbers 2
N
=
{
0
,
2
,
4
,
6
,...
}
? In the previous example the difference was
just one element. But in this example, there seem to be twice as many natural numbers as there are even
natural numbers. Surely, the cardinality of
N
must be larger than that of 2
N
since
N
contains all of the odd
natural numbers as well. Though it might seem to be a more difficult task, let us attempt to find a bijection
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 Spring '08
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 Natural number, Countable set

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