# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 3

*Sets*

Mathematical structures like numbers are described as sets. A *set* is a collection of distinct objects which are called the *elements* of the set. By distinct we mean that it is clear which objects are considered to be equal and which are considered to be different. The *containment* of an element to a set is expressed by

the noncontainment by

For every element, exactly one of these possibilities holds. For example, we have
and
.
An important principle for sets is the *principle of extensionality*, i.e. a set is determined by the elements it contains, beyond that it bears no further information. In particular, two sets coincide if they contain the same elements.

The set which does not contain any element is called the *empty set* and is denoted by

A set is called a *subset* of a set if every element from does also belong to . For this relation we write
(some people write
for this). One also says that the *inclusion*
holds. For the number sets, the inclusions

hold. The subset relation
is a statement using for all, as it makes a claim about all elements from . If we want to show
,
then we have to show for an arbitrary element
that also the containment
holds. In order to show this, we are only allowed to use the property
.
For us, sets will be either number sets or sets constructed from such number sets. A set is called *finite* if its elements may be counted by the natural numbers for a certain
.
In this case, the number is called the *number* (or the *cardinality*) of the set.

*Possible descriptions for sets*

There are several ways to describe a set. The easiest one is to just list the elements of the set, here the order of the listing is not important. For finite sets this is possible, however, for infinite sets one has to describe a "construction rule“ for the elements.

The most important set given by an infinite listing is the set of natural numbers

Here a certain set of numbers is described by a list of starting elements in the hope that this makes it clear which numbers belong to the set. An important point is that is not a set of certain digits, but the set of values represented by these digits or sequences of digits. For a natural number there are many possibilities to represent it, the decimal expansion is just one of them.

We discuss now the description of sets by properties. Let denote a given set. In there are certain elements which fulfil a certain property (a predicate) or not. Hence, for the property we have within the subset consisting of all the elements from which fulfil this property. We write for this subset given by

This only works for such properties for which the statement is well-defined for every . If one introduces such a subset then one gives a name to it which often reflects the name of the property, like

For the sets occurring in mathematics, a multitude of mathematical properties is relevant and therefore there is a multitude of relevant subsets. But also in the sets of everyday life like the set of the students in a course, there are many important properties which determine certain subsets, like

The set itself is also given by a property, since

*Set operations*

Similar to the connection of statements to get new statements, there are operations to make new sets from old ones. The most important operations are the following.^{[1]}

*Union**Intersection**Difference set*

For these operations to make sense, the sets need to be subsets of a common basic set. This ensures that we are talking about the same elements. Quite often this basic set is not mentioned explicitly and has to be understood from the context. A special case of the difference set is the *complement* of a subset
in a given base set , also denoted as

If two sets have an empty intersection, meaning
,
we also say that they are *disjoint*.

*Product set*

We want to describe within set theory the arithmetic operations on the number sets mentioned above like addition and multiplication. For the addition (say on ), two natural numbers and are added to yield another natural number, namely . The set of pairs constitute the product set and the adding is interpreted as a mapping on the product set.

We define.^{[2]}

Suppose that two sets and are given. Then the set

is called the *product set*

*Cartesian product*) of the sets.

The elements of a product set are called *pairs* and denoted by . Here the ordering is essential. The product set consists of all pair combinations, where in the first *component* there is an element of the first set and in the second component there is an element of the second set. Two pairs are equal if and only if they are equal in both components.

If one of the sets is empty, then so is the product set. If both sets are finite, say the first with elements and the second with elements, then the product set has elements. It is also possible to form the product set of more than two sets.

Let be the set of all first names, and be the set of all last names. Then

is the set of all names. The elements of this set are in pair notation , and . From a name, one can deduce easily the first name and the last name by looking at the first or the second component. Even if all first names and all last names do really occur, not every combination of a first name and a last name does occur. For the product set, all possible combinations are allowed.

For a product set it is also possible that both sets are equal. Then one has to be careful not to confuse the components.

A chess board (meaning the set of squares of a chess board where a chess piece may stand) is the product set . Every square is a pair, e.g. . Because the two component sets are different, one may write instead of pair notation simply . This notation is the starting point to describe chess positions, and complete chess games.

The product set is thought of as a plane, one denotes it also by . The product set is a set of lattice points.

Let denote a circle (the circumference), and let be a line segment. The circle is a subset of a plane , and the line segment is a subset of a line , so that for the product sets, we have the relation

The product set is the three-dimensional space, and the product set is the surface of a cylinder.

*Mappings*

Suppose that a particle is moving in space. This process is described by assigning to every point in time the point in space where the particle is at that time. The running of a computer program that uses altogether memory units can be described by saying for every computing step (which means the execution of a program line) what the content of the memory units is. So to the number counting the computing steps, we assign the tuple

In an election, every voter has to decide for exactly one party ( or for not voting). The temperature profile on the earth is described by assigning to every point in time and every point on the surface its temperature. Such and many other situations are captured by the concept of a mapping.

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

If a mapping
is given, then is called the *domain* (or domain of definition) of the map and is called the *codomain* (or *target range*) of the map. For an element
,
the element

is called the *value* of at the *place* (or *argument*) .

Two mappings
and
are equal if and only if their domains coincide, their codomains coincide and if for all
the equality
in
holds. So the equality of mappings is reduced to the equalities of elements in a set. Mappings are also called *functions*. However, we will usually reserve the term *function* for mappings where the codomain is a number set like the real numbers .

For every set , the mapping

which sends every element to itself, is called the *identity* (on ). We denote it by
. For another set and a fixed element
,
the mapping

which sends every element
to the *constant value* , is called the *constant mapping*
(with value ).
It is usually again denoted by .^{[3]}

There are several ways to describe a mapping, like value table, bar chart, pie chart, arrow diagram, the graph of the mapping. In mathematics, a mapping is most often given by a mapping rule, which allows computing the values of the mapping for every argument. Such rules are e.g.
(from to )
, , etc. In the sciences and in sociology also *empirical functions* are important which describe real movements or developments. But also for such functions, one wants to know whether they can be described (approximated) in mathematical manner.

*Injective and surjective mappings*

Let and denote sets, and let

be a
mapping.
Then is called *injective*, if for two different elements
,
also
and

If we want to show that a certain mapping is injective then we may show the following: For any two elements and fulfilling the condition , we can deduce that . This is often easier to show than the statement that implies .

Let and denote sets, and let

be a
mapping.
Then is called *surjective*, if for every
,
there exists at least one element
,
such that

We consider a football game as the mapping which assigns, to every goal of team , the corresponding goal scorer. Suppose that there are no own goals and no changes. The goals of are numbered by . Then we have a mapping

given by

The injectivity of means that every player has scored at most one goal, the surjectivity means that every player has scored at least one goal.

Let denote the set of all (living or dead) people. We study the mapping

which assigns to every person his or her (biological) mother. This is well-defined, as every person has a uniquely determined mother. This mapping is not injective, since there exists different people (brothers and sisters) with the same mother. It is also not surjective, since not every person is a mother of somebody.

The mapping

is neither injective nor surjective. It is not injective, because the different numbers and are both sent to . It is not surjective, because only nonnegative elements are in the image (a negative number does not have a real square root). The mapping

is injective, but not surjective. The injectivity can be seen as follows: If , then one number is larger, say

But then also , and in particular . The mapping

is not injective, but surjective, since every nonnegative real number has a square root. The mapping

is injective and surjective.

Let and denote sets and suppose that

is a
mapping.
Then is called *bijective* if is
injective
as well as

The question, whether a mapping has the properties of being injective or surjective, can be understood looking at the equation

(in the two variables and ). The surjectivity means that for every there exists at least one solution

for this equation, the injectivity means that for every there exist at most one solution for this equation, and the bijectivity means that for every there exists exactly one solution for this equation. Hence, surjectivity means the existence of solutions, injectivity means the uniqueness of solutions. Both questions are everywhere in mathematics and they also can be interpreted as surjectivity or injectivity of suitable mappings.

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

So we have

where the left-hand side is defined by the right-hand side. If both mappings are given by functional expressions, then the composition is realized by plugging in the first term into the variable of the second term (and to simplify the expression if possible).

*Footnotes*

- ↑ It is easy to memorize the symbols: the for union looks like u. The intersection is written as . The corresponding logical operations or, and have the analogous form and , respectively.
- ↑ In mathematics, definitions are usually presented as such and get a number so that it is easy to refer to them. The definition contains the description of a situation where only concepts are used which have been defined before. In this situation, a new concept together with a name for it is introduced. This name is printed in a certain font, typically in
*italic*. The new concept can be formulated without the new name, the new name is an abbreviation for the new concept. Quite often, the concepts depend on parameters, like the product set depends on its component sets. The names are often chosen arbitrarily, the meaning of the word within the mathematical context can be understood only via the explicit definition and not via its meaning in everyday life. - ↑ Hilbert has said that the art of denotation in mathematics is to use the same symbol for different things.

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