Math 100
Finite Sets
Martin H. Weissman
1.
Finite Sets
Suppose that
A
and
B
are sets, and
f
:
A
→
B
is a function.
•
f
is called injective if the following sentence is true:
If
x, y
∈
A
, and
f
(
x
) =
f
(
y
), then
x
=
y
.
•
f
is called surjective if the following sentence is true:
If
b
∈
B
, then there exists
a
∈
A
, such that
f
(
a
) =
b
.
•
f
is called bijective if
f
is injective and surjective.
We will often use an informal definition of finiteness, motivated by counting:
Definition 1.
Suppose that
S
is a set, and
n
∈
N
. Then, we say that
S
is a finite set with
n
elements, if it is possible to order and name the elements of
S
with symbols
s
1
up to
s
n
:
S
=
{
s
1
, . . . , s
n
}
.
Many of the proofs in these notes are difficult to read. Here is an important piece of advice:
when reading the proofs in these notes, draw “arrow diagrams”, in order to understand what
is going on with the functions.
2.
Injective Functions
Using the above informal definition, we can prove the following:
Proposition 2.
Suppose that
A
and
B
are finite sets, and
m
= #
A
, and
n
= #
B
. Then,
there exists an injective function from
A
to
B
if and only if
m
≤
n
.
Proof.
First, we suppose that
m
≤
n
, and we will prove that there exists an injective function
from
A
to
B
.
Since
A
is a finite set with
m
elements, we can name its elements:
A
=
{
a
1
, . . . , a
m
}
.
Since
B
is a finite set with
n
elements, we can name its elements:
B
=
{
b
1
, . . . , b
n
}
. Then, one may define a function
f
from
A
to
B
by:
f
(
a
i
) =
b
i
,
for all
i
∈
N
,
such that 1
≤
i
≤
m.
Given two elements
a
i
, a
j
∈
A
, if
f
(
a
i
) =
f
(
a
j
), then
b
i
=
b
j
. It follows that
i
=
j
, so
a
i
=
a
j
.
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 Fall '08
 Weissman
 Sets, elements, Finite set

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