It is true that thousands of years ago, humanity began to conceive the idea of numbers and began to do calculations, until 2500 years ago, some philosophers began to wonder about the numbers themselves, apart from
practical usage. They invented the axiomatic deductive method and established thus mathematics as an exact science. Philosophy (love of wisdom) had thus come across a field, where you can penetrate to the truth.
Every cheeky assertion or assumption must be proved or can be disproved. But it was only in the twentieth century, that mathematicians succeeded in bringing the by then
scattered results under one roof and onto common foundations. This is sometimes referred to as the modern mathematics; Adler [01]. Since then, the lectures for
beginners at university are not about numbers, it's all about propositions and how to handle them. We have even to pretend to be creatures, who are utterly ignorant of numbers, but who are about to
discover operations with certain things.

During your first weeks at university this may be bothersome for you. But make the best of it and internalize this mode of thinking, if possible, well in advance.

When I was in this situation, there was a blind student among us. Instead of doing nothing after leaving school, he had worked hard, so as not to fall behind on account of his handicap. Once in a while,
lecturers would ask him, if he is able to follow an argument. Unerringly, he used to answer: "I can see very clearly". Quite often, the majority of us was irritated, because we were in the dark but didn't
have the guts to ask. Most lecturers like it, when you come forward with questions.

With "P := Today is Sunday", we define P to mean a whole sentence, in order to make our work easy and more lucid. For doing so, we use another abbreviation := (conceptualization). Also =: is in use; the colon is next to what is being defined. We learn here also, that a proposition needs not absolutely to be true or false. It may depend on circumstances.

A compound proposition is an assembly of statements which are connected by sentential connectives (logical connectives) like the conjunction ∧ (AND-operation) and the disjunction ∨ (OR-operation).

If the proposition P implies the proposition Q, (implication), we write P ⇒ Q (or Q ⇐ P). Hold both, P ⇒ Q and Q ⇒ P, we write P ⇔ Q (equivalence).

So as to save even more space, ink and energy when writing statements, we use the energy-saving quantifiers ∀ ("for all") and ∃ ("there exists") that indicate the scope of a term to which it is attached. (Sometimes we may come across ∃! := "there exist exactly one").

P | Q | ¬P | ¬Q | P ∨ ¬Q | ¬P ∨ ¬Q | (P ∨ ¬Q) ∧ ( ¬P ∨ ¬Q) |

T | T | F | F | T | F | F |

T | F | F | T | T | T | T |

F | T | T | F | F | T | F |

F | F | T | T | T | T | T |

So far we have not yet defined any number system, but we know that it won't be long; also we know from school, that we are going to handle sets of numbers.

In mathematics, if we make use of a concept that hasn't been defined yet, we say that we use the concept naively. We still use the different number concepts naively. As far as the concept of a set
is concerned, we leave it at the naive use, because the theory of sets is an axiomatic deductive theory in itself
(Halmos [26, 26a], Ebbinghaus [13]), to which you can devote yourself fully, if you are interested, when you have climbed the
astonishing stairs of the basics and can have a look around in the big reception hall of the mathematical building.

By a *set* S we mean an object that is a collection of *well-distinguished* objects, which need not be arranged and which are called the *elements* of S.

With { } we symbolize a container which we call the *empty set*, a set, which doesn't contain any element. For the *non-empty* sets M := {a, b, a,} and {b, a} =: N,
M = N is a true proposition, because every element counts only once, and the order is not relevant. We have a ∈ N (a is an element of N) and c ∉ M (c is not an element of M).

We use Venn diagrams in order to visualize operations with sets, as shown in Abb 1 to 6. (We don't use diagrams to define or prove anything).

With C := A ∩ B = {x | x ∈ A ∧ x ∈ B} we assign C the intersection of the sets A and B (Abb 1).

If, as result of this operation, C is not empty, it contains all those elements that A and B have in common (Abb 1).

If C is empty, we say that A and B are disjoint (Abb 3).

If however A ∩ B = B holds, we write B ⊂ A (inclusion) and call B a subset of A (Abb 2) or we say B is contained in A.

If both B ⊂ A and A ⊂ B hold, then A = B holds.

Since { } contains no element which is not contained in every other set, we have { } ⊂ M for every set M.

In order to obtain the union of two sets A and B, we empty them into one bag, discard with duplicates and write A ∪ B (Abb 6).
Therefore A ∪ A = {x | x ∈ A ∨ x ∈ A} = A (not 2A).

If a set B doesn't want to have something in common with a set A, it removes its intersection with A and calls himself from now on B - A, the
relative complement of A (Abb 4).

The set A ⊕ B := (A - B) ∪ (B - A) = (A ∪ B) - (A ∩ B)
is called the *disjoint union* of A and B; see (Abb 5).

∅ := { } (the empty set)

We have: ∅ ⊂

With |M| we denote the cardinality of the set M. For a set M having a finite number n of elements, we write |M| = n, for a set M having an infinite number of elements, we write |M| = ∞. (Cantor

If for sets M and A, A ⊂ M holds, we call B := M - A the complement of A (in M). For example, the even numbers form the complement of the uneven numbers (in the set of the integers

We can also form sets whose elements are sets. But be careful to avoid sets that contain themselves, or similar nonsense. The power set P(M) of a set M is the set of all subsets of M, including Ø and M. If M := {a, b, c}, then P(M) = {Ø, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, M}. For a finite set M,

|P(M)| = 2^{|M|}.

Let A_{1}, A_{2}, ... ,A_{n} be sets. The product of these sets is written A_{1} x A_{2} x ... x A_{n}. this is a new set, also called the *Cartesian product* of the A_{i} and the elements of this new set are called n-tupels which are of the form (a_{1}, a_{2}, ... ,a_{n}), where a_{i} ∈ A_{i}. If A_{1}
has n_{1} elements, there are n_{1} possibilities to occupy the first place in the n-tupel. For the second position there are n_{2} possibilities ... , for the last position there are n_{n}
possibilities. It is now easy to see, that |A_{1} x A_{2} x ... x A_{n}| = |A_{1}| ⋅ |
A_{2}| ⋅ ... ⋅ |A_{n}|.

An example for A_{i} = A_{j} ist **R**^{2} := **R** x **R** =: **C**. The elements of **C** are the complex numbers (a, b). The first position in this 2-tupel is occupied by the real part, the second position by the imaginary part.
The number (a, b) is also written as a + ib.

**R** x **R** is a fine example for the fact, that a set becomes only "useful", when we provide it with operations and axioms. In the shape of **C**, **R**^{2} constitutes the complex numbers in
the Gaussian number plane, in analytical geometry of the plane it is the set of coordinates of the points of the plane and in a 2-dimensional vector space it serves as set of elements of the space.

The Cartesian product A x B depicted in Abb 7 has 20 elements. Each subset of it represents a relation. Let R be the relation {(e, 3), (c, 2), (a, 4), then we have for example (b, 1) ∉ R but (e, 3) ∈ R. Let A := the set of the inhabitants of Kremsmünster, B := the set of the inhabitants of Buxtehude, and L the relationen "is in love with". Then, as far as I know, there is only one element in this relation A x B. Franz from Kremsmünster is in love with Marion from Buxtehude. From this relation,{(Franz, Marion)} we are not able to tell, if Marion is in love with Franz too (symmetry), or whether Marion loves Karl and therefore, Franz also loves Karl (transitivity) ore if Franz loves himself (reflexivity). Properties of this kind can only be found in Cartesian products A x A, like for example in

**Equivalence relation on a set and partion of a set**

If S is a set and R is a relation on S, then R is said to be an equivalence relation ⇔

a) x ∈ S ⇒ xRx

b) x, y ∈ S and xRy ⇒ yRx

c) x, y, z ∈ S and xRy and yRz ⇒ xRz

The properties a, b, c are called the *reflexive*, *symmetric* and *transitive* properties, respectively. A popular label for equivalence relations is ∼. An equivalence relation always goes
along with a partition (decomposition) of the supporting set S into disjoint subsets, the equivalence classes [a]_{∼} = {x ∈ S | x∼a} of ∼. The set of equivalence
classes C|_{∼} = {[x]_{∼} | x ∈ S} has thus the same elements as S, only grouped into equivalence classes.
Example: the set of the even and the set of the odd numbers are disjoint and their union gives
the set of the integers. In this relation, the even numbers have nothing to do with the uneven numbers and vice versa (e_{i}∼e_{j}, e_{i}≁o_{j}, o_{i}∼o_{j}, o_{i}≁e_{j}).

**Partial orders**

are relations which are reflexive, antisymmetric and transitive and for aRb we write a ≼ b. For example in the power set P(M) = {{ }, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, M} of the set M = {a, b, c},
the elements of the sequence ({ }, {b}, {b, c}, {a, b, c}) are comparable in this way. In the Hasse diagram in Abb 8 you can detect other sequences.

are relations which are irreflexive, antisymmetric and transitive and for aRb we write a ≺ b. In this case, any two elements of the supporting set are comparable.

A group (G, ∘) is a nonempty set G together with a binary operation ∘ defined on G such that the following axioms are satisfied:

G1.

G2.

G3.

(G, ∘) is

A Ring (R, +, ⋅) is a nonempty set R together with two binary operations + and ⋅ (

R1. (R, +) is an abelian Group

R2. Multiplication is associative

R3. The following distributive laws are satisfied:

For all a, b, c ∈ R

a ⋅ (b + c) = a ⋅ b + a ⋅ c, (b + c) ⋅ a = b ⋅ a + c ⋅ a

A ring in which the multiplication is commutative is a

A Field is a nontrivial, commutative ring with