Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/List of definitions
The set which does not contain any element is called the empty set, denoted by
Let and denote sets. is called a subset
of if every element of is also an element of .For sets und , we call
the intersection
of the two sets.For sets und , we call
the union
of the sets.Suppose that two sets and are given. Then the set
Let and denote sets. A mapping from to is given by assigning, to every element of the set , exactly one element of the set . The unique element which is assigned to , is denoted by . For the mapping as a whole, we write
Let and denote sets, and let
be a mapping. Then is called injective, if for two different elements , also and
are different.Let and denote sets, and let
be a mapping. Then is called surjective, if for every , there exists at least one element , such that
Let and denote sets and suppose that
is a mapping. Then is called bijective if is injective as well as
surjective.Let denote a bijective mapping. Then the mapping
which sends every element to the uniquely determined element with ,
is called the inverse mapping of .Let and denote sets, let
and
be mappings. Then the mapping
is called the composition of the mappings
and .A set is called a field if there are two binary operations (called addition and multiplication)
and two different elements , which fulfill the following properties.
- Axioms for the addition:
- Law of associativity: holds for all .
- Law of commutativity: holds for all .
- is the neutral element of the addition, i.e. holds for all .
- Existence of the negative: For every , there exists an element with .
- Axioms of the multiplication:
- Law of associativity: holds for all .
- Law of commutativity: holds for all .
- is the neutral element for the multiplication, i.e. holds for all .
- Existence of the inverse: For every with , there exists an element such that .
- Law of distributivity: holds for all .
For a natural number , one puts
Let and denote natural numbers with . Then
A field is called an ordered field, if there is a relation (larger than) between the elements of , fulfilling the following properties ( means or ).
- For two elements , we have either or or .
- From and , one may deduce (for any ).
- implies (for any ).
- From and , one may deduce (for any ).
Let be an ordered field. is called Archimedean, if the following Archimedean axiom holds, i.e. if for every there exists a natural number such that
For real numbers , , we call
- the closed interval.
- the open interval.
- the half-open interval (closed on the right).
- the half-open interval (closed on the left).
For a real number , the modulus is defined in the following way.
The set with and , with componentwise addition and the multiplication defined by
is called the field of complex numbers. We denote it by
The degree of a nonzero polynomial
with
is .For polynomials , , the function
where is the complement of the zeroes
of , is called a rational function.Let denote a positive real number. The Heron-sequence, with the positive initial value , is defined recursively by
Let denote a real sequence, and let . We say that the sequence converges to , if the following property holds.
For every positive , , there exists some , such that for all , the estimate
holds.
If this condition is fulfilled, then is called the limit of the sequence. For this we write
A subset of the real numbers is called bounded, if there exist real numbers such that
.A real sequence is called a Cauchy sequence, if the following condition holds.
For every , there exists an , such that for all , the estimate
Let be a real sequence. For any strictly increasing mapping , the sequence
An ordered field is called complete or completely ordered, if every Cauchy sequence in
converges.A sequence of closed intervals
in is called (a sequence of) nested intervals, if holds for all , and if the sequence of the lengths of the intervals, i.e.
A real sequence is said to tend to , if for every , there exists some , such that
A real sequence is said to tend to , if for every , there exists some . such that
Let be a sequence of real numbers. The series is the sequence of the partial sums
If the sequence converges, then we say that the series converges. In this case, we write also
for its limit,
and this limit is called the sum of the series.Let be a subset,
a function, and a point. We say that is continuous in the point , if for every , there exists a , such that for all fulfilling , the estimate holds. We say that continuous, if it is continuous in every point
Let denote a subset and a point. Let
be a function. Then is called limit of in , if for every there exists some such that for all fulfilling
the estimate
holds. In this case, we write
Let be a sequence of real numbers and another real number. Then the series
For two series and of real numbers, the series
The real number
For a positive real number , the exponential function for the base is defined as
For a positive real number , , the logarithm to base of is defined by
The function defined for by
The function defined for by
The function