Transcript Lecture 4

Lecture 4
Infinite Cardinals
Some Philosophy: What is “2”?
 Definition 1: 2 = 1+1. This actually needs the
definition of “1” and the definition of the “+”
operation.
 Definition 2: Start with the concept of “two
apples”, and remove all aspects of a single apple,
e.g. redness, taste, etc.. You’ll be left with the
number “2”. This definition is a bit problematic.
 Definition 3: 2 = The class of all sets of size 2 (this
is indeed a very large class)
 Definition 4: 2 = {0,1}, where 1 = {0} and 0 = {}.
Note: 2 is a particular set of size 2.
Some History: What is “n”?
 Historically, people could not count beyond some
(relatively small) finite number, e.g. 10.
 A (large) number “n” did not have a name, but
people could access it by having a bag with n
stones.
 If a shepherd wants to make sure the number of
sheep was n, he matches the sheep with the stones.
 Thus, two sets have the same size if there is a
bijection (one-to-one correspondence) between the
elements of the sets..
Some Definitions
 The size of a set A is less than or equal to that
of B, written A  B iff there is an injective (oneto-one) function f:A  B.
 The size of a set A equals the size of the set B,
written A  B iff there is a bijective (one-to-one
and onto) function f:A  B.
 We say that the set A is equipotent (or equinumerous) with B.
 Note: If A is finite and has n elements, we can
take the size of A = n. However, the size of an
infinite set A is yet to be defined.
Some Simple Facts about 
 Obviously: A  B  A  B.
 The relation  on sets is:
Reflexive, i.e. for all sets A, A  A.
 Symmetric, i.e. for all sets A and B;
A  B  B  A.
 Transitive, i.e. for all sets A,B and C;
(A  B and B  C)  A  C.
 The above properties are easy to prove.
 Thus,  is an equivalence relation on the
class of all sets.
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Some Simple Facts about 
 The relation  on sets is:
Reflexive, i.e. for all sets A, A  A.
 Transitive, i.e. for all sets A,B and C;
(A  B and B  C)  A  C.
 Antisymmetric, i.e. for all sets A,B;
(A  B and B  A)  A  B.
 The first two properties are easy to
prove, the third constitutes an important
theorem…
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Cantor–Bernstein–Schroeder Theorem:
 For all sets A and B;
(A  B and B  A)  A  B.
 Proof Outline: We have two injective
functions f :A  B and g :B  A. We use
these to construct a bijection h :A  B.
 The idea is to find a suitable subset C  A,
such that
 f (a) if a  C
 (see
h(a)   1
 g (a) if
http://en.wikipedia.org/wiki/Cantor%E2%80%93Bernstein%E2%80%93Schroeder_theorem
)
a C
The following infinite sets are countably
infinite (i.e. equipotent with N):
 The set of even numbers
 The set of prime numbers (or in general any
infinite subset of N)
 The set of rational numbers
 The set of algebraic numbers (see
 The set of computable reals (see
 The set computer programs
 The set of computer files
http://en.wikipedia.org/wiki/Algebraic_numbers
http://en.wikipedia.org/wiki/Computable_number
)
)
Is every infinite set equipotent with N?
 Answer: No! the set of reals R is larger than N.
 Proof: Clearly N  R. N  R means that there is a
bijection f: N  R, i.e. a listing of all reals of the
form x1,x2,x3,….
 We can then construct a real number y distinct from
any infinite list of real numbers by letting:
the ith digit of y  the ith digit of xi
Picture
Change all digits of the diagonal
.4 8 2 0 8 2 1 9 0 …
.7 0 7 3 6 9 6 3 9 …
.1 9 6 3 2 9 4 9 2 …
.0 7 4 9 3 8 9 5 1 …
.9 3 2 8 9 4 2 9 0 …
.4 6 8 5 3 2 8 0 0 …
.3 6 8 0 5 6 2 1 8 …
.7 5 3 7 8 0 8 1 3 …
.0 8 7 4 2 8 6 8 0 …
... … … … … … … … … …
Picture
to get the number: .517003321…
.5 8 2 0 8 2 1 9 0 …
.7 1 7 3 6 9 6 3 9 …
.1 9 7 3 2 9 4 9 2 …
.0 7 4 0 3 8 9 5 1 …
.9 3 2 8 0 4 2 9 0 …
.4 6 8 5 3 3 8 0 0 …
.3 6 8 0 5 6 3 1 8 …
.7 5 3 7 8 0 8 2 3 …
.0 8 7 4 2 8 6 8 1 …
... … … … … … … … … …
In general: Cantor’s Theorem
 For every set A, its power set defined by
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P(A) = {X: XA} is larger than A.
Proof: Clearly A  P(A).
If A  P(A) , then there is a bijection
f: A  P(A).
However, the subset B of A defined by:
B = {aA: af(a)} is not covered by f.
If it were, i.e. B = f(a), for some a, then:
aB  af(a)  aB, a contradiction.
The Continuum Hypothesis
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We have infinitely many infinities.
We call these ‫א‬0, ‫א‬1, ‫א‬2,… (the alephs)
These are the infinite cardinal numbers
‫א‬0 (called aleph_0) denotes the size of N.
We say: |N| = ‫א‬0
Question: Is |R| = ‫א‬1?
(This is the so-called Continuum Hypothesis)
 Answer: Our Mathematics is too weak to
decide this question (assuming it’s consistent)!
Cardinal Arithmetic
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Definition: Let |A| = , and |B| = . We define:
 +  = |A  B| (if they are disjoint)
   = |A  B|
 = |BA|, where BA is the set of all functions from
B to A.
Note: These definition generalize the arithmetic of
natural numbers.
Facts: If one of  and  is infinite, then:
 +  =    = max{,}
If   , then  = 2 > 
Thank you for listening.
Wafik