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.
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…
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
P(A) = {X: XA} 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 = {aA: af(a)} is not covered by f.
If it were, i.e. B = f(a), for some a, then:
aB af(a) aB, a contradiction.
The Continuum Hypothesis
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
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