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