LECTURE 24: CARDINALITY (
§
10.1-10.2)
We have too many high sounding words, and too few actions that correspond with them.
Abigail
Adams
1.
Cardinality Revisited
We have used
|
A
|
to denote the number of elements in
A
. But for infinite sets this is not precise.
Instead, we will measure the size of sets in a more precise way.
Notice that
A
=
{
a, b, c, d
}
and
B
=
{
1
,
2
,
3
,
4
}
have the same number of elements. One way to see
this is to create a bijection between them. We can do a similar thing with infinite sets.
Definition 1.
Two sets
A
and
B
are said to have the
same cardinality
if there is a bijection
f
:
A
→
B
.
Note that
f
-
1
:
B
→
A
is also a bijection, so this relation is symmetric. We write
|
A
|
=
|
B
|
.
If
A
and
B
have no bijections between them, we write
|
A
| 6
=
|
B
|
.
Theorem 2.
The relation of having the same cardinality is an equivalence relation.
Proof.
First we show this relation is reflexive. Let
A
be a set. We want a bijection from
A
to itself.
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- Spring '11
- Smith
- Sets, Natural number, Bijection, Denumerable
-
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