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Chapter 2
From Finite to Uncountable Sets
A considerable amount of the material offered in this chapter is a review of termi
nology and results that were covered in MAT108. Our brief visit allows us to go
beyond some of what we saw and to build a deeper understanding of some of the
material for which a revisit would be bene
¿
cial.
2.1
Some Review of Functions
We have just seen how the concept of function gives precise meaning for binary
operations that form part of the needed structure for a
¿
eld. The other “big” use of
function that was seen in MAT108 was with de
¿
ning “set size” or cardinality. For
precise meaning of what constitutes set size, we need functions with two additional
properties.
De
¿
nition 2.1.1
Let A and B be nonempty sets and f
:
A
±²
B. Then
1. f is
onetoone
, written f
:
A
1
±
1
B, if and only if
±
1
x
²±
1
y
1
z
²±±
x
³
z
²
+
f
F
±
y
³
z
²
+
f
"
x
³
y
² ³
2. f is
onto
, written f
:
A
±
B, if and only if
±
1
y
y
+
B
"
±
2
x
x
+
A
F
±
x
³
y
²
+
f
²²³
3. f is a
onetoone correspondence
, written f
:
A
1
±
1
±
B, if and only if f is
onetoone and onto.
49
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CHAPTER 2. FROM FINITE TO UNCOUNTABLE SETS
Remark 2.1.2
In terms of our other de
¿
nitions, f
:
A
±²
B is onto if and only if
rng
±
f
²
³
def
´
y
+
B
:
±
2
x
²±
x
+
A
F
±
x
³
y
²
+
f
²
µ ³
B
which is equivalent to f
[
A
]
³
B.
In the next example, the
¿
rst part is shown for completeness and to remind the
reader about how that part of the argument that something is a function can be
proved. As a matter of general practice, as long as we are looking at basic functions
that result in simple algebraic combinations of variables, you can assume that was
is given in that form in a function on either its implied domain or on a domain that
is speci
¿
ed.
Example 2.1.3
For f
³
t
x
³
x
1
± ¶
x
¶
u
+
U
·
U
:
±
1
´
x
´
1
}
, prove that
f
:
±
±
1
³
1
²
1
±
1
±
U
.
(a) By de
¿
nition, f
l
U
·
U
±
i.e., f is a relation from
±
±
1
³
1
²
to
U
.
Now suppose that x
+
±
±
1
³
1
²
.Then
¶
x
¶
´
1
from which it follows that
1
± ¶
x
¶ /³
0
. Hence,
±
1
± ¶
x
¶
²
±
1
+
U
± ´
0
µ
and y
³
x
¸
±
1
± ¶
x
¶
²
±
1
+
U
because multiplication is a binary operation on
U
. Since x was arbitrary, we
have shown that
±
1
x
x
+
±
±
1
³
1
²
"
±
2
y
y
+
U
F
±
x
³
y
²
+
f
²²
±
i.e.,
dom
±
f
²
³
±
±
1
³
1
²
.
Suppose that
±
x
³
y
²
+
f
F
±
x
³)²
+
f. Thenu
³
x
¸
±
1
± ¶
x
¶
²
±
1
³
)
because multiplication is singlevalued on
U
·
U
.S
i
n
c
ex
³
u, and
)
were
arbitrary,
±
1
x
1
u
1
)²±±
x
³
u
²
+
f
F
±
x
+
f
"
u
³
)²
±
i.e., f is singlevalued.
Because f is a singlevalued relation from
±
±
1
³
1
²
to
U
whose domain
is
±
±
1
³
1
²
, we conclude that f
:
±
±
1
³
1
²
²
U
.
2.1. SOME REVIEW OF FUNCTIONS
51
(b) Suppose that f
±
x
1
²
±
f
±
x
2
²
±
i.e., x
1
³
x
2
+
±
²
1
³
1
²
and
x
1
1
² ³
x
1
³
±
x
2
1
² ³
x
2
³
.
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This note was uploaded on 09/24/2008 for the course MAT 215 taught by Professor Pandhirapande during the Fall '08 term at Princeton.
 Fall '08
 PANDHIRAPANDE
 Sets

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