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MAT 3749Introduction to Analysis
Section 1.3 Part I
Countable Sets
http://myhome.spu.edu/lauw
Goals
Review and Renew the concept of functions•How to show that a function is an One-
to-one function (Injection)
•How to show that a function is an Onto function (Surjection)
Countable and Uncountable Sets
References
Section 1.3 Howland, Appendices A-C
You Know a Lot About Functions
You are supposed to know a lot… Domain, Range, Codomain Inverse Functions One-to-one, Onto Functions Composite Functions
Is this a Function? (I)
X Y
Is this a Function? (II)
X Y
One-to-One Functions
: is if
for each , there is at most on
1-1
e
such that (
(inject
)
ive)f X Y
y Y x X
f x y
X Y
Equivalent Criteria
1 2
1 2 1 2
For , ,
if ( ) ( ) then
x x X
f x f x x x
X Y
Example 1Determine if the given function is injective. Prove your answer. :
( ) 3 1
f
f n n
Z Z
Onto Functions
onto (surjective: is if
the range of
)
is .
f X Y
f Y
X Y
Equivalent Criteria
, such that ( )y Y x X f x y
X Y
Example 2Determine if the given function is surjective. Prove your answer. :
( ) 3 1
f
f n n
Z Z
Counting Problems…
X Y
?
X Y
Counting Problems…
X Y
?
X Y
Bijections
: is if it is both 1b -1 and onto.ijectivef X Y
Inverse Functions
1
If : is
then its inverse function : exists
and i
bijec
s also bijecti e
ive
v
t
.
f X Y
f Y X
Equivalent Sets
Example 3
The set of odd integers (O) and even integers (E) are equivalent.
Plan:
1. Define a function from O to E.
2. Show that the function is well defined.
3. Show that the function is bijective.
Countable Sets
Remark
Theorem
Analysis
Proof
Proof
Proof
Corollary (HW)
Theorem
Proof Outline
1 2 3 4 5
1 1 1 1 1
1 2 3 4
2 2 2 2
1 2 3
3 3 3
1 2
4 4
1
5
Proof Outline
2 4
2 2
1 2 3 4 5
1 1 1 1 1
1 3
2 2
1 2
3 3
1
4
1
5
3
3
2
4
Proof Outline
2 4
2 2
1 2 3 4 5
1 1 1 1 1
1 3
2 2
1 2
3 3
1
4
1
5
3
3
2
4