LECTURE 24: CARDINALITY (
§
10.110.2)
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.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 Smith
 Sets, Natural number, Bijection, Denumerable

Click to edit the document details