1. Function Hubert Chan (Chapter 2.1, 2.2) [O1 Abstract Concepts]
[O2 Proof Techniques]

2. What you will learn… Z+ = {1,2,3…..} E+ = {2,4,6,8,…} Real numbers
Z+ and E+ are countable sets
Rational numbers are countable
Real numbers are uncountable
To prove that they have different kinds of infinity, we need the concept of function

3. What is a Function? Let us consider some mathematical functions
f(x) = 3x-2
square-area(x) = x2
Function represented by table
Function represented by graph
Price (HK$) 5
5.5 6
6.5 7
7.5
Demand (no.)
5000
4700
4500
4350
4250
4200

4. Functions and Sets [O1] f(i)=j means put ball i into bin j.
Let A, B be sets. A function f from A to B is an assignment of exactly one element of B to each element of A.
We write f : A  B
We can also write f(a) = b where aA and bB
A function is also called a mapping.
Example: （Labeled balls into labeled bins）
A function f represented by labeled balls into labeled bins from A (the set of balls) to B (the set of bins) is defined as follows:
f(i)=j means put ball i into bin j.
Hence the function is putting balls into bins and each ball is put into exactly one bin.

5. Basic Terminology Given a function f : A  B ( f maps A to B ),
A is the domain of f (‘balls’)
B is the codomain of f (‘bins’)
If f(a) = b,
b is the image of a.
The range of f is the set of all images of elements of A.
{ b | b  B and (  a (f (a) = b) ) } (‘non-empty bins’) A B
Example:
Let f : Z  Z and f(x) = x2, where Z = set of integers
Domain = Z, codomain = Z, range = Z ?

6. Injective Functions (One-to-one)
A function f is injective, iff for every distinct x, y in the domain, f(x)  f(y).
“Different inputs imply different outputs”
Examples of injective function:
f: Z  Z: f(x)=2x, f(x)=x3,
balls into bins function such that each bin has at most one ball.
Examples of not injective function:
f: Z  Z: f(x)=x4 A B

7. Surjective Functions (Onto)
A function f from A to B is called surjective,
iff for all b  B, there exists an element a  A such that f(a) = b.
“Every element in the codomain is the output of some input”
Examples of surjective function:
f: Z  Z: f(x)=x+2
balls into bins function such that each bin has at least one ball.
Examples of not surjective function
f: Z  Z: f(x)=2x A B

8. Bijective Functions
A function f that is both injective and surjective is called a bijection.
Examples:
f: Z  Z: f(x)=x, f(x)=x+2
balls into bins function such that each bin has exactly one ball.
which means the number of balls = the number of bins if there are finite number of balls and bins. A B

9. Examples
Determine whether the following functions are injection, surjection, or bijection?
f1 : Z  Z, f1(x) = x2
Not injection, since f(-1) = f(1) = 1.
Not surjection, since there is no integer x such that
x2 = -1.
f2 : Z  Z, f2(x) = x + 1
Injection, since x+1  y+1 implies x  y.
Surjection, since f(x) = y  x + 1 = y  x = y – 1.
Bijection, since it is both one-to-one and onto.

10. Different Kinds of Infinity
Two sets A and B have the same cardinality
iff there exists a bijection f:A→B
Example:
Z+ = {1,2,3,4,…..}
E+ = {2,4,6,8,…}
Does E+ Z+ imply |E+| < |Z+|?
Bijection f : Z+→E+ such that f(x)=2x
NO!

11. Different Kinds of Infinity
f : Z+→E+ such that f(x)=2x is
Injective since if x1, x2  Z+ are different, f(x1) ≠ f(x2)
Surjective since for any y  E+, there exists x=y/2 such that x  Z+ and f(x)=y
f(x) is a bijection
Z+ and E+ have the same cardinality: ….
↓ ↓ ↓ ↓ ↓ … ….

12. Countable Sets Definition. A set S is countable if either:
(i) the set S is finite, or
(ii) there is a bijection from the set S to Z+.
Examples of Countable Sets
Z+ (positive integers) is countable.
Set of all integers
Even numbers

13. Some Results Countable and uncountable:
All subsets of a countable set are countable. Proof?
If there is a injection from a set A to another set B
If A is uncountable, then B is uncountable.
If B is countable, then A is countable, too.

14. Q (rational numbers) is countable [O2]
Simple case: Q+ is countable.
Each element of Q+ can be represented by a/b, where a and b are relatively-prime integers and a,b>0. b a 1 2 3 4 …
1/1
1/2
1/3
1/4
2/1
2/2
2/3
3/1
3/2
4/1 Q+

15. Q (rational numbers) is countable
Simple case: Q+ is countable.
Each element of Q+ can be represented by a/b, where a and b are relatively-prime integers and a,b>0. b a 1 2 3 4 …
1/1
1/2
1/3
1/4
2/1
2/3
3/1
3/2
4/1
Removing duplicates 1 2 4 6 Q+ Z+ 3 7 5 8
Mapping to
integers 9

16. Question: All infinite sets are countable?
Examples:
Z+ = {1,2,3,4,…..}
E+ = {2,4,6,8,…}
Q = rational numbers
R = real numbers

17. Question: Do Z+ and P(Z+) have the same cardinality?
They are all infinite sets.
But they have different kinds of infinities.
P(Z+) contains “more” elements and is uncountable.
Theorem:
There is NO bijection mapping from any non-empty set S to P(S)
P(S) has a larger cardinality than S

18. There is NO bijection mapping any non-empty set S to P(S)
Theorem:
There is NO bijection mapping any non-empty set S to P(S)
Proof: (proof by cases and contradiction)
Case 1: If S is a finite set. Then |P(S)| = 2|S| > |S|.
Case 2: If S is a infinite set.
Assume there is a bijection f: S → P(S). Then for each xS, f(x)P(S).
Define , then A  P(S).
Since f is also a surjection, for each element X in P(S), there exists x in S such that f(x)=X. Hence there exists an element a in S such that f(a)=A.
If , then condition* of A is not satisfied, Contradiction.
If , then condition* of A is satisfied, Contradiction.
Hence, such a bijection does not exist.
condition*

19. R (real numbers) is uncountable
Intuition: P(Z+) is uncountable and is a “subset” of R.
There is a injection from P(Z+) to R:
Input: a subset of Z+, S={s1, s2, s3,…}
Output: a real number X in [0,1] such that
X=0.x1x2x3x4… where xi=1 if iS and 0 if iS
Examples:
f({1,3,4,7,10}) =
f({2,5,6,8,9}) =
f(Ø)=0, f(Z+)= …
R has at least the cardinality as P(Z+)
R is uncountable!

20. Axiom of Choice
(Something sounds trivial but has profound implication later)