Functions and infinite sets

1
Functions and infinite sets

• (Approx 2-3 lectures. Initial sections adapted
  from slides for a course by Michael P. Frank)

2
Reading relevant chapters of any book on
Discrete Maths. For example, Rosen 5th ed., 1.8
3
Functions

• From calculus, you know the concept of a
  real-valued function f, which assigns to each
  number x?R one particular value yf(x), where
  y?R.
• Example f defined by the rule f(x)x2
• Roughly, functions say the so-and-so of
• Functions are also called operations, mappings,
  etc.

4
Functions

• To understand functions more precisely, one needs
  the mathematical notion of a set
• We assume you are familiar with naïve set
  theory (as opposed to axiomatic set theory).
• In a nutshell

5
Reminder of main set concepts

• ?, ?, ?, ?, ?,
• , ?, ?, ?, ?, ?, etc.
• a,b,... (def. of a set by enumeration)x
  P(x) (def. by set builder notation)
• x?S, S?T, ST, S?T.
• P(S) (power set of S),
• A?B (Cartesian product of A and B)

6
Reminder of main set concepts

• Important sets of numbers N, Z, Q, R
• A relation on A is a subset of AxA. E.g.,
• on, N, the relation lt is (0,1),(0,2), (1,2),
• Set equality proof techniques
• E.g., to prove AB, prove each of
• A?B
• B?A

7
Function Formal Definition

• A function f from (or mapping) A to B (fA?B)
  is an assignment of exactly one element f(x)?B to
  each element x?A.
• Generalisations
• Functions of n arguments f (A1 x A2... x An) ?
  B.
• A partial (non-total) function f assigns zero or
  one elements of B to each element x?A.

8
Functions precisely

• We can represent a function fA?B as a set of
  ordered pairs f (a,f(a)) a?A.
• This makes f a relation between A and Bf is a
  subset of A x B. But functions are special
• for every a?A, there is at least one pair (a,b).
  Formally ?a?A?b?B((a,b)?f)
• for every a?A, there is at most one pair (a,b).
  Formally ??a,b,c((a,b)?f ? (a,c)?f ? b?c)
• A relation over numbers can be represent as a set
  of points on a plane. (A point is a pair (x,y).)
• A function is then a curve (set of points), with
  only one y for each x.

9
Useful diagrams

• Functions can be represented graphically in
  several ways

A
B
f


f




y

a
b




x
A
Bipartite Graph
B
Plot
Like Venn diagrams
10
Functions that youve seen before

• A set S over universe U can be viewed as a
  function from the elements of U to

11
Still More Functions

• A set S over universe U can be viewed as a
  function from the elements of U to
• T, F, saying for each element of U
  whether it is in S. (This is called the
  characteristic function of S)
• Suppose U0,1,2,3,4. Then
• S1,3? S(0)S(2)S(4)F, S(1)S(3)T.

12
Still More Functions

• A set operator, such as ? or ?, can be viewed as
  a function from to

13
Still More Functions

• A set operator such as ? or ? can be viewed as a
  function from (ordered) pairs of sets, to
  sets.
• Example ?((1,3,3,4)) 3

14
A new notation

• YX is the set F of all possible functions f X?Y.
• Thus, f ? YX is another way of saying f X?Y.
• (This notation is especially appropriate, because
  for finite X, Y, we have F YX. )

15
Some Function Terminology

• If fA?B, and f(a)b (where a?A b?B), then we
  say
• A is the domain of f.
• B is the codomain of f.
• b is the image of a under f.
• a is a pre-image of b under f.
• In general, b may have more than 1 pre-image.
• The range R?B of f is Rb ?a f(a)b .

We also saythe signatureof f is A?B.
16
Range versus Codomain

• The range of a function may not be its whole
  codomain.
• The codomain is the set that the function is
  declared to map all domain values into.
• The range is the particular set of values in the
  codomain that the function actually maps elements
  of the domain to.

17
Choosing the right (co)domain

• Consider the function f such that f (x) 100/x
• Is f a (total) function from Int to R?
• f is a partial function from Int to R
• f is a (total) function from Int-0 to R
• Consider the function g such that g(x) vx
• Is g a (total) function from R to R?
• g is a total function from R to RxR
• e.g. g(4) (2,-2)

18
Images of Sets under Functions

• Given fA?B, and S?A,
• The image of S under f is the set of all images
  (under f) of the elements of S. f(S) ? f(s)
  s?S ? b ?s?S f(s)b.
• The range of f equalsthe image (under f) of ...

19
Images of Sets under Functions

• Given fA?B, and S?A,
• The image of S under f is the set of all images
  (under f) of the elements of S. f(S) ? f(s)
  s?S ? b ?s?S f(s)b.
• The range of f equalsthe image (under f) of fs
  domain.

20
One-to-One Functions

• A function is one-to-one (1-1), or injective, or
  an injection, iff every element of its range has
  only 1 pre-image.
• Formally given fA?B,f is injective ?
  (??x,y x?y ? f(x)?f(y)).
• In other words only one element of the domain is
  mapped to any given one element of the range.
• In this case, domain range have same
  cardinality. What about codomain?

21

• Codomain may be larger.

22
One-to-One Illustration

• Are these relations one-to-one functions?

23
One-to-One Illustration

• Are these relations one-to-one functions?

One-to-one
24
One-to-One Illustration

• Are these relations one-to-one functions?

Not one-to-one
One-to-one
25
One-to-One Illustration

• Are these relations one-to-one functions?

Not even a function!
Not one-to-one
One-to-one
26
Sufficient Conditions for 1-1ness

• For functions f over numbers, we say
• f is strictly increasing iff xgty ? f(x)gtf(y) for
  all x,y in domain
• f is strictly decreasing iff xgty ? f(x)ltf(y) for
  all x,y in domain
• If f is either strictly increasing or strictly
  decreasing, then f must be one-to-one.
• Does the converse hold?

27
Onto (Surjective) Functions

• A function fA?B is onto or surjective or a
  surjection iff its range is equal to its codomain
  (?b?B, ?a?A f(a)b).
• Consider country of birth of A?B,where
  Apeople, Bcountries. Is this a function? Is
  it an injection? Is it a surjection?

28
Onto (Surjective) Functions

• A function fA?B is onto or surjective or a
  surjection iff its range is equal to its codomain
  
• Consider country of birth of A?B,where
  Apeople, Bcountries. Is this a function? Yes
  (always 1 c.o.b.) Is it an injection? No (many
  have same c.o.b.) Is it a surjection? Probably
  yes ..

29
Onto (Surjective) Functions

• A function fA?B is onto or surjective or a
  surjection iff its range is equal to its codomain
• In predicate logic

30
Onto (Surjective) Functions

• A function fA?B is onto or surjective or a
  surjection iff its range is equal to its
  codomain.
• In predicate logic
• ?b?B?a?A f(a)b

31
Onto (Surjective) Functions

• A function fA?B is onto or surjective or a
  surjection iff its range is equal to its codomain
  (?b?B?a?A f(a)b).
• E.g., for domain codomain Z, the function
  ?x.x1 is injective and surjective.

32

• Claim if fZ?Z and f(x) x1 then
• f is 1-to-1 and also onto.
• (Z is the set
  of all integers)
• Proof that f is onto Consider any arbitrary
  element a of Z. We have f(a-1)a, where a ? Z.
• Proof that f is 1-to-1 Suppose f(u)f(w)a. In
  other words, u1a and w1a. It follows that uw.

33
Onto/surjective functions

• Are these functions onto their depicted
  co-domains?

34
Onto/surjective functions

• Are these functions onto?

35
Onto/surjective functions

• Are these functions onto?

onto
not onto
onto
not onto
36
1-1/injective functions

• Are these functions 1-1?

onto
not onto
onto
not onto
37
1-1/injective functions

• Are these functions 1-1?

not 1-1onto
not 1-1not onto
1-1onto
1-1not onto
38
Bijections

• A function is said to be a one-to-one
  correspondence, or a bijection iff it is both
  one-to-one and onto.

39
Two terminologies for talking about functions

• injection one-to-one
• surjection onto
• bijection one-to-one correspondence
• 3 12

40
Bijections

• For bijections fA?B, there exists a function
  that is the inverse of f, written f ?1 B?A
• Intuitively, this is the function that undoes
  everything that f does
• Formally, its the unique function such that
• ...

41
Bijections

• For bijections fA?B, there exists an inverse of
  f, written f ?1 B?A
• Intuitively, this is the function that undoes
  everything that f does
• Formally, its the unique function such that
• (the identity function on A)

42
Bijections

• Example 1 Let f Z?Z be defined as f(x) x1.
  What is f?1 ?
• Example 2 Let g Z?N be defined as g(x) x.
  What is g?1 ?

43
Bijections

• Example 1 Let f Z?Z be defined as f(x)x1.
  What is f?1 ?
• f?1 is the function (lets call it h) h Z?Z
  defined as h(x)x-1.
• Proof

h(f(x)) (x1)-1 x
44
Bijections

• Example 2 Let g Z?N be defined as g(x)x.
  What is g?1 ?
• This was a trick question there is no such
  function, since g is not a bijectionThere is no
  function h such that h(x)x and h(x)?x
• (NB There is a relation h for which this is
  true.)

45
Cardinality (informal)

• The cardinality of a finite set is its number of
  elements
• E.g., card(a,b,c) card(e,f,g) 3
• Note for finite sets X and Y, card(X)card(Y)
  if and only if there exists a bijection between X
  and Y

46
infinity

• This is straightforward if a set has 0 or 1 or 2
  or n (any natural number) elements
• But what if the set has more elements than that?
  Some examples
• The set of all natural numbers itself
• The set of all even natural numbers
• L(1) 1, 11,111,1111,11111, etc.
• L(01) 01,001,011,0001,0011, etc

47
The Diagonalisation Method

• Georg Cantor (1873) Whats the size of an
  infinite set?
• E.g., is card(L(1)) card(L(01))?
• Both are infinite
• But is one larger than the other?
• Cantors idea
• The size (cardinality) of a set should not depend
  on the identity of its elements
• Two finite sets A and B have the same size if we
  can pair the elements of A with elements of B
• Formally there exists a bijection between A and B

48
Correspondences (Contd)

• Example Let
• N be the set of pos. natural numbers 1, 2, 3,
  
• E the set of even pos. natural numbers 2, 4, 6,
  
• Using Cantors definition of size, we can show
  that N and E have the same size
• Bijection (!) f (n) 2n
• Intuitively, E is smaller than N, but
• Pairing each element of N with its corresponding
  element in E is possible,
• So we declare these two set to be the same size
• This even though E ? N (E is a real subset of
  N )

49
Countable sets

• A set X is finite if it has n elements, for some
  n in N.
• A set is countable if either
• It is finite or
• It has the same size as N, the natural numbers
• For example,
• N is countable, and so are all its subsets
• E is countable
• 0,1,2,3 is countable
• ? is countable
• How about supersets of N ?

50
An Even Stranger Example

• Let Q be the set of positive rational numbers
• Q m/n m,n ? N
• Just like E, the set Q has the same size as N !
• We show this giving a bijection from Q to N
• Q is thus countable
• One way is to enumerate (i.e., to list) Qs
  elements.
• Pair the first element of Q with 1 (first elt. of
  N )
• And so on, making sure every member of Qappears
  only once in the list

51
An Even Stranger Example (Contd)

• To build a list with the elements of Q
• make inf. matrix with all positive rational
  numbers
• i -th row contains all numbers with numerator i
• j -th column has all numbers with denominator j
• i /j is in i -th row and j -th column

52
An Even Stranger Example (Contd)

• Now we turn the previous matrix into a list
• A bad way begin list with first row
• Since rows are infinite, we will never get to 2nd
  row!

53
An Even Stranger Example (Contd)

• Instead, we list the elements along diagonals

We should, however, eliminate repeated elements
54
An Even Stranger Example (Contd)

• We list elements along diagonals w/o repetitions

?1/1, 2/1, 1/2, 3/1, 1/3, ?
55
Uncountable sets

• Some sets have no correspondence with N
• These sets are simply too big!
• They are not countable we say uncountable
• Theorem
• The set of real numbers between 0 and 1(e.g.,
  0.244, 0.3141592323....) is uncountableCall this
  set R0,1
• (Some sets are even larger. Serious set
  theory is all about theorems that concern
  infinite sets. Most of this is irrelevant for
  this course.)

56

• Theorem R0,1 gt N. Proof strategy
• R0,1 gtN. Suppose R0,1 N and derive a
  contradiction Each member of R0,1 can be written
  as a zero followed by a dot and a countable
  sequence of digits. Suppose there existed a
  complete enumeration of R, (using whatever order)
  lte1,e2,e3,...gt.

57

• showing how an arbitrary list might start
• e1. 0.0000000000000000000000....
• e2. 0.0100000000000000000000....
• e3. 0.8200000000000000000000....
• e4. 0.1710000000000000000000....
• ...

58
Now construct a Real number n thats not in the
enumeration

• ns first digit (after the dot) e1s first
  digit 1
• ns second digit e2s second digit 1 ...
• General ns i-th difit e-is i-th digit 1
• ?i n differs from e-i in its i-th digit
• Contradiction lte1,e2,e3,...gt is not a (complete)
  enumeration after all. QED

59

• This proof of the non-countability of the set of
  Real numbers is known as Cantors diagonalisation
  argument
• It proved to be the start of a large new area of
  set theory, involving the cardinalities of
  infinite sets

60
The Russell Paradox

• For example, read Rosen 5th ed., 1.6especially
  ex. 30 on p. 86

61
Basic Set Notations

• Set enumeration a, b, c
• and set-builder xP(x).
• ? relation, and the empty set ?.
• Set relations , ?, ?, ?, ?, ?, etc.
• Venn diagrams.
• Cardinality S and infinite sets N, Z, R.
• Power sets P(S).

62
Axiomatic set theory

• Various axioms, e.g., saying that the union of a
  set of sets is a set the intersection of a set
  of sets is a set etc.
• One key axiom Given a Predicate P, one can
  construct a set. It consists of all those
  elements x such that P(x) is true.
• But, the resulting theory turns out to be
  logically inconsistent!
• This means, there exists set theory proposition p
  such that both p and ?p follow logically from
  the axioms of the theory!
• ? The conjunction of the axioms is a
  contradiction
• This makes the theory is fundamentally
  uninteresting, because any possible statement in
  it can be (very trivially) proven!

63
Prove

• Theorem Given a contradiction, any statement can
  be proven

64
Prove

• Theorem Given a contradiction, any statement can
  be proven
• Proof Let your contradiction be p ?p
  (the assumption is youve proven it before)
• Suppose you want to prove q
• (p ?p) --gt q is a tautology of propositional
  logic(Check truth table of the formula, given p
  ?p is false)
• Youve proven p ?p
• q follows with Modus Ponens. Note that q is
  arbitrary!

65
This version of Set Theory is inconsistent

• Russells paradox
• Consider the set that corresponds with the
  predicate x ? x
• S x x?x .
• Now ask is S?S?

66
Russells paradox

• Let S x x?x . Is S?S?
• If S?S, then S is one of those x for which x?x.
  In other words, S?SWith Proof by Contradiction,
  we have S?S
• If S?S, then S is not one of those x for which
  x?x. In other words, S?SWith Proof by
  Contradiction, we have S?S
• We conclude that both S?S nor S?S
• Paradox!

67

• To avoid inconsistency, set theory had to somehow
  change

Bertrand Russell1872-1970
68
One technique to avoid the problem

• Given a set S and a predicate P, construct a new
  set, consisting of those elements x of S such
  that P(x) is true.
• Youll seen this technique in use when we get to
  the programming language Haskell, where we can
  write
• x x ? 1.. , even x, but not x even x.

69
Another technique to avoid the problem

• Russells paradox arises from the fact that we
  can write funny things like x?x (or x?x, for
  that matter). One solutionforbid such
  expressions using types.
• Youll seen this technique in use as well
  Haskells use of typing.

70
Our focus computability

• We shall not worry about saving set theory from
  paradoxes like Russells
• Instead, we shall use the Russell paradox in a
  different setting
• But first we need to talk about Formal Languages,
  Haskell, and Turing Machines

71
(No Transcript)