Download SET THEORY A set is a collection (family) of distinct and well

SET THEORY
A set is a collection (family) of distinct and well defined objects. More precisely,
a collection of well defined and distinct objects is a set if it belongs itself to such a
collection. The objects of a set are called elements, and a subcollection of a set is a
set as well and called a subset.
We are not developing details of such an approach. It is left to lectures on the
foundation of mathematics and set theory. What we shall need in this course is the
set N of it positive integers or natural numbers which is assumed to be a family of well
defined objects (the natural numbers are denoted by 1,2,3,...) having some additional
structure:
(a) Each natural number n has a successor (that is denoted by n + 1) which also
belongs to N,
(b) the element 1 belongs to N and is not the successor of any element in N,
(c) if two elements n and m in N have the same successor, then n and m represent
the same element in N,
(d) A subset of N which contains 1 and with n also its successor, must be equal
to N.
These are known as the Peano’s axioms of the number system. Note that d) is the
basis for mathematical induction!
All sets we will consider are well defined in terms of this basic set N. To summarize:
If S is a set, its defining objects are called elements and s ∈ S means that s is an
element of the set S. Likewise we say s belongs to S or write S 3 s and so on... If
we want to specify which elements are in S we shall write
S = {s1 , s2 , ...},
where the brackets are used to enclose the elements, which are all assumed to be
pairwise different and well defined. Hence we write
N = {1, 2, 3, ...}
such that 2 is the successor of 1, 3 that one of 2, .... We also use the notation
S = {s : s has properties A,B,...}.
If t ∈ S and s ∈ S we also write s, t ∈ S. Is s is not an element of S we write
s 6∈ S, and similarly for more than two elements. If the two letters s and t are used
for the same element of S we shall write s = t. We say s is equal to t. This is an
equivalence relation as explained below:
• s=s
• s=t⇔t=s
• s = t and t = r ⇒ s = r.
It is possible to form chains of such relations.
If S is a set, a set S0 is called a subset of S if every element of S0 is also an element
of S, formally if
∀s : s ∈ S0 =⇒ s ∈ S.
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We shall write S0 ⊂ S or S ⊃ S0 . Two subsets S0 and S1 are called equal if they
contain exactly the same elements of S, we write S0 = S1 . The set S0c = {s ∈ S : s 6∈
S0 } is called the complement of S0 (in S). Often, it is written as S0c = S \ S0 . The
subset of S which does not contain any element of S is called the emptyset. It will
be denoted by
∅ = {} ⊂ S.
Note that {∅} =
6 ∅. Here we use the symbol 6= to denote ¬ =, so
{∅} =
6 ∅ ⇐⇒ ¬({∅} = ∅).
If E and F are subsets of S, then
E ∪ F = {s ∈ S : s ∈ E or s ∈ F }
is the union of E and F . There is a close connection to the logical notion of alternative:
Let s ∈ S. The statement A is that s ∈ E and the statement B is that s ∈ F . Then
A or B means the statement that s ∈ E ∪ F .
If E and F are subsets of S, then
E ∩ F = {s ∈ S : s ∈ E and s ∈ F }
is the intersection of E and F . There is also a close connection to the logical notion
of conjunction: Let s ∈ S. The statement A is that s ∈ E and the statement B is
that s ∈ F . Then A and B means the statement that s ∈ E ∩ F .
The difference of two subsets E and F of F is defined by
E \ F := {s ∈ S : s ∈ E and s 6∈ F }.
One can verify that
E \ F = E ∩ F c.
We learned that every subcollection of a set is a set itself. This is so because it is
an object in the collection of all subcollection of this set. The set of all subsets of a
set S is denoted by 2S and called the power set of S. Similar one shows that for two
sets S and T the collection
S × T = {(s, t) : s ∈ S, t ∈ T }
is a set, called the product of S and T .
We apply these constructions of new sets to define
Z = N ∪ {0} ∪ {−n : n ∈ N}
as the set of integers and
Q = {(p, q) : p ∈ Z, q ∈ N, gcd(p, q) = 1}
to be the set of rational numbers. (gcd= greates common divisor). Thus Z and Q
are sets as well and N ⊂ Z ⊂ Q by canonical identification (!!).
We assume that all basic operations on Q are permitted, such as multiplication, division (except by 0 of course), addition, subtraction and the
order structure
(p, q) ≤ (p0 , q 0 ) ⇐⇒ pq 0 ≤ p0 q.
For better handling, we write (p, q) = pq .
Reference for these properties: Ross: Section 1.3, page 13.
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FUNCTIONS
Like the notion of a set, the terminology of a function is basic in mathematics
(and logical thinking). A function describes a functional or algorithmic dependence
between two quantities. In mathematics quantities are elements and need to be
members of sets. So let S and S ∗ be two sets (which may be equal). A function f
from S to S ∗ is a rule assigning to each element s ∈ S a value f (s) ∈ S ∗ . In terms of
axiomatic set theory: A function is a subset
F ⊂ S × S ∗ = {(s, s∗ ) : s ∈ S and s∗ ∈ S ∗ }
such that each s ∈ S occurs exactly once. We then define f (s) = s∗ if and only if
(s, s∗ ) ∈ F . F is called the graph of the function f . If S0 ⊂ S then a function f
from S to S ∗ with domain S0 is a function f from S0 to S ∗ . The subset {s∗ ∈ S ∗ :
∃s ∈ S0 and (s, s∗ ) ∈ F } is called the range of f . A usual description of a function
f : S0 → S ∗ is to identify f with an element of
∗
S0S = {(s∗s )s∈S0 : ∀s ∈ S0 : s∗s ∈ S ∗ }.
With this description in mind one shortly writes
f : S0 → S ∗
to denote a function with domain S0 and range contained in S ∗ . Note that f (s) is
not a function but a function value. Other terminology for functions: map, mapping,
transformation, operator. The elements s ∈ S0 and s∗ ∈ S ∗ are called the variables of
the function, the first one the “independent” the second one the “dependent” variable.
We write s∗ = f (s), and later, when using the real line, y = f (x).
Two functions f and g are called equal, written as f = g, if they are defined on
the same domain S0 with ranges in the same set S ∗ and
∀s ∈ S0 :
f (s) = g(s),
the last equality as elements of the set S ∗ .
We can map the integers Z into the rational numbers Q defining
f (z) = (z, 1).
Clearly, this map is injective which means that for z 6= z 0 it follows that f (z) 6= f (z 0 ).
In this case one says that Z is embedded into Q.
We say that two sets S and S 0 have the same cardinality if there is a map
f : S → S0
which is injective (s 6= s0 ∈ S =⇒ f (s) 6= f (s0 ) ∈ S 0 ) and onto (∀t ∈ S 0 ∃s ∈ S =⇒
f (s) = t). The set {1, ..., n} has cardinality n, the emptyset cardinality 0 and N
has a cardinality which is called countable or denumerable. So any set which is an
ıisomorphisc image (that is the range of an injective map) of N is countable.
Theorem 0.1. The rational numbers are countable.
Proof. We need to construct a map f : N → Q which is injective and onto. In order
to do so we use the counting method by counter-diagonals.
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Fix a natural number q. Then the rational numbers of the form p/q with gcd(p, q) =
1 are countable (each such rational has a successor and a smallest element can be
set to be 1/q.) Denote this enumeration by qn (n ∈ N). Then define f (1) = 0,
f (2) = 1 = 11 , f (3) = 1/2 = 21 , f (4) = 2 = 12 , and so on.
q.e.d.
Theorem 0.2. A product set A × B = {(s, t) : s ∈ A, t ∈ B} of denumerable sets is
denumerable.
PROJECT 1: Let M and F be two sets. Suppose that for each f ∈ F there is an
associated subset M (f ) ⊂ M given. Find conditions under which there is an injective
map H : F → M such that for each f ∈ F
H(f ) ∈ M (f ).
You may check literature under the name of marriage problem. Discuss this problem and, if possible, find a solution.
EQUIVALENCE RELATIONS
Let S be a set. A relation s ∼ t for elements s, t ∈ S is a function f which is defined
on S × S = {(s, t) : s, t ∈ S} and has a range in S ∗ = {Y es, N o} with f (s, t) = Y es
if and only if s ∼ t. A relation is called an equivalence relation if it has the following
properties
• s∼s
• s∼t⇔t∼s
• s ∼ t and t ∼ r =⇒ s ∼ r.
In general, equality of elements of sets is an equivalence relation. Also, as is used
quite often, transforming equations is an equivalence relation: Let (1) and (2) be two
equations, each of them can be transformed into the other (meaning that if (1) is
true if and only if (2) holds). Then this transformation rule is an equivalence relation
on the set of all equations. It means that (1) can be shown to be true if one finds a
transformed equation which is known to be true.
This method of proof is mostly used in the proofs in calculus.
Example. Let S = Z denote the integers. Then we define the relation x ∼ y if and
only x-y is divisible by 2 (this means that ∃p ∈ Z such that x − y = 2p). We show
that this defines an equivalence relation:
• Since we know that x = x we can transform this equation into x − x = 0. Also
we know that 2 divides 0, hence we have by definition that x ∼ x.
• Let x ∼ y. Then we know that 2 divides x − y, so x − y = 2p for some p ∈ Z.
Multiplying this equation by −1, which is an admissible transformation as
explained above, we obtain y − x = 2(−p), hence 2 divides y − x and so y ∼ x.
• Let x ∼ y and y ∼ z. Then by definition, 2 divides x − y and y − z, so
x − y = 2p and y − z = 2q with p, q ∈ Z. Adding both equations (which is an
admissible transformation) yields
x − z = x − y + y − z = 2p + 2q = 2(p + q)
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with p + q ∈ Z. Therefore 2 divides x − z and x ∼ z.