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Transcript
```MAT 3749 1.3 Part I Handout
Just For Fun……ction?
One-to-One Functions
f : X  Y is 1-1 (injective) if for each
y  Y , there is at most one x  X such that
f  x  y .
Equivalent Criteria
For x1 , x2  X ,
if f  x1   f  x2  then x1  x2
Last Edit: 4/30/2017 11:51 AM
Example 1
Determine if the given function is injective.
f : Z  Z , f (n)  3n  1
Onto Functions
f : X  Y is onto (surjective) if the range
of f is Y .
Equivalent Criteria
y  Y ,  x  X such that f ( x)  y
2
Example 2
Determine if the given function is surjective.
f : Z  Z , f (n)  3n  1
Bijections
f : X  Y is bijective if f is both 1-1 and onto.
Inverse Functions
If f : X  Y is bijective, then its inverse function f 1 : Y  X exists and is also
bijective.
3
Equivalent Sets
Two sets A and B are equivalent if there is a 1-1 function from A onto B .
Example 3
The set of odd and even integers are equivalent.
Analysis
Proof
Countable Sets
A set A is countable if it is either finite or N is equivalent to A .
Remark
If an infinite set A is countable, then we can list its element as a sequence
A  a1 , a2 , a3 ,  .
4
Theorem
The interval  0,1 is not countable.
Analysis
5
Proof
Suppose  0,1 is countable. Since  0,1 is not a finite set, the elements of  0,1 can be
listed as a sequence an n 1 . For each of the real number an , we can represent it by its
infinite decimal expansion. Thus, the sequence is
 a1  0.a11a12 a13
a  0.a a a
21 22 23
,
*  2
 a3  0.a31a32 a33

where each aij 0,1, 2, ,9 ij  N .

Construct a real number x  0.x1 x2 x3
such that n N ,
5 ann  5
.
xn  
4 ann  5
Then,
1. x   0,1 .
2. n N , xn  ann  x  an which is a contradiction to x   0,1 .
Thus,  0,1 is not countable.
Corollary
The reals are equivalent to  0,1 , and are hence uncountable.
HW
6
Theorem
The set Q of rationals is countable.
Proof Outline
1. List all the positive rational numbers as follows:
1
1
1
2

1
3
1
4

1
5

2
1
3
1
2
2
3
2
2
3
3
3

4
1
5
1
4
2
2
4
2. Form a sequence an n 1 by following the arrows and omitting any number that have

3. The sequence 0, a1 , a1 , a2 , a2 , a3 , a3 ,
list all the rational numbers.
7
Classwork (Family of Sets)
In this classwork, we are going to review the concept and notations of Family of Sets.
Below,  a, b  and  a, b represent open and closed intervals respectively.
Recall
Let A and B be two sets.
A  B  x | x  A or x  B and A  B  x | x  A and x  B .
10
1. We know
a
n 1
n
 a1  a2 
 a10 . Let An be sets for n  1, 2,...,10 . What do you
10
think the notation
An represents?
n 1
2. Let An   n, n  1 for nN . Find
 1 1
3. Let Bn    ,  for nN . Find
 n n
10
An .
n 1
10
Bn .
n 1
8
Variations
10
An for I  1, 2,...,10
An can be written as
n 1
nI
4. Let An   n, n  1 for nN and F  1, 2,8,9 . Find
An .
nF
 1 1
5. Let Bn    ,  for nN and F  1, 2,8,9 . Find
 n n
Bn .
nF
9
Here are the formal definitions for the union and intersection of families of sets.
Definition
Let F be a set and  A n nF be a family of sets. Then,
An   x | x  An for some n  F  and
nF
6. Let Cy   2,sin y  for y 
An   x | x  An for all n  F  .
nF
.
(a) Make an educational guess on the explicit interval form of
C y . No need to
y