MAT 3749
Introduction 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-toone 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
f : X  Y is 1-1 (inject ive) if
for each y  Y , there is at m ost on e x  X
such that f ( x )  y
X
Y
Equivalent Criteria
For x1 , x 2  X ,
if f ( x1 )  f ( x 2 ) then x1  x 2
X
Y
Example 1
Determine if the given function is injective. Prove your
answer.
f :Z  Z
f (n )  3n  1
Onto Functions
f : X  Y is onto (surjective ) if
the range of f is Y .
X
Y
Equivalent Criteria
 y  Y ,  x  X such that f ( x )  y
X
Y
Example 2
Determine if the given function is surjective. Prove
your answer.
f :Z  Z
f (n )  3n  1
Counting Problems…
Y
X
?
X

Y
Counting Problems…
Y
X
?
X

Y
Bijections
f : X  Y is b ijective if it is both 1 -1 and onto.
Inverse Functions
If f : X  Y is bijec t ive
then its inverse function f
and i s also bijecti v e .
1
: Y  X exists
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
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
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
