Set (mathematics)

The amount is one of the most important and basic concepts of mathematics. You pick up individual elements to a lot together ( in mathematics in particular figures, but also, for example, in statistics, the people being tested in a sample, people of a year, people with high blood pressure as a suspected risk factor for diseases ) in the context of set theory. A lot must not contain element (there is only one set with no elements, the " empty set "). In the description of a set is all about the question of which elements are included in it. It will not be asked whether an element is included more than once, or if there is an order among elements.

• 2.1 Equality of sets and extensionality 2.1.1 equality
• 2.1.2 extensionality

Concept and notation of quantities

The term amount goes back to Bernard Bolzano and Georg Cantor. In Bolzano's manuscripts from the years 1830 to 1848 it says: " epitomes now, bey which on the way its parts are connected to each other, should not be taken, where is thus Everything we distinguish them determines when only part of their [ own ] are determined, deserve expressing this nature will to be referred to with its own acquisitions. In the absence of another suitable word I allow myself the word amount for this purpose need to; ". Cantor described a lot of " naive " as a summary of certain, well- differentiated objects of our intuition or our thought to a whole. The objects of the set are called elements of the set. Neither the term " amount" nor the term " element " are defined in the mathematical sense; they are defined not as or in axioms. The modern set theory and thus a large part of mathematics is based on the Zermelo -Fraenkel axioms, Neumann - Bernays - Gödel axioms or other axiom systems. We have a natural, intuitively correct understanding of quantities; leads, however, the term " the set of all sets that do not contain themselves as an element " to a contradiction of Russell's antinomy; as well as " the set of all sets ".

An illustration of the concept of quantity, which is attributed to Richard Dedekind, is the image of a sack which certain ( identifiable as individuals ) contains things. Useful is this notion, for example, for the empty set: an empty sack. The empty set is not " nothing ", but the content of a container that does not contain the provisions for it as the content of things. The " container " itself refers only to the certain to be counted variety and type of elements.

Finite amounts ( especially if they have relatively few elements ) by enumerating its elements ( enumerative set notation ) can be specified, such as M = { blue, yellow, red }, where it as I said does not depend on an order or whether an item more is referred to as a time. That is, it is true, for example, { blue, red, yellow } = { blue, yellow, red } = { blue, blue, yellow, red }.

Often it is inconvenient or (for infinite sets ) impossible to enumerate the elements of a set. In these cases, there is a different notation, in which the elements of a set are defined by a property, for example M = { x | x is a base color }.

Furthermore Dedekind coined the synonym of the system to which he summarized elements. This name is still partially usual, it is called a " set of vectors " also briefly a vector system.

Other spellings

Other notations for quantities can be regarded as abbreviations for the intensional notation:

• The enumerative notation M = { blue, yellow, red } can be used as a shorthand for the more cumbersome notation M = { x | x = x = blue or yellow or red = x } be understood.
• In the notation ellipsis only a few items are listed as examples, such as: M = {3, 6, 9, 12, ..., 96, 99 }. It is only usable when the Education Act from these examples or the context is clear. Here obviously the amount is meant the intensional as M = { to x | x is a number divisible by 3 between 1 and 100 } can write. This notation is frequently used for infinite sets. Describes as G = {4, 6, 8, 10, ... } is the set of even positive integers greater than 2
• New volumes can also form by set operations, such as from A and B the intersection. This can be written as M = { intensional x | x is in A and x is in B}.
• Further there is also the definition of inductive amounts, wherein at least a base member is explicitly specified, and then one or more rules, such as an additional element may be derived from an element. Thus, the above quantity G can also be described by

Cardinality

For finite sets, the cardinality (or cardinality) is equal to the number of elements of the set; is a natural number including zero. The concept can be generalized also to infinite sets; it turns out that two infinite sets need not be equally powerful. The cardinality of a set is with in general, sometimes also listed with.

Basic relations between quantities

The things that are included in an amount, called elements. If an object is member of a set, so you write it formally. The negation ( is not an element of ) one writes as:. Historically, the symbol goes back to the Greek letter ε as the initial letter of εστί ( ESTI it is).

Equality of sets and extensionality

Equality

Two sets are called equal if they contain the same elements.

This definition refers to the extensionality and thus the fundamental property of sets. Formal:

In fact, a lot needs but are usually described intensionally. This means that there is a statement form specified ( with an object variable that should have a well-defined set of definitions ), so if and only if is true. But then you write:

There are many different forms of expression to describe this is any set. The question of whether two given statement forms and the same amount describe is by no means trivial. On the contrary: Many problems in mathematics can be formulated in this form: " Are and the same amount? "

Many equality proofs use the equivalence.

Extensionality

If two sets have the same elements, so they are identical. On the way, as the assignment of the elements is described with the amounts, it is not to. The characteristic quantities property that it does not depend on the type of description, we call their extensionality (from the Latin extensio = expansion; concerns the size of the content ).

Infinite quantities have but mostly " intensional " (descriptive set notation ) are described (from the Latin intensio = voltage; concerns the characteristics of the content ). That is: A set is described by a certain condition or property that all elements of the set (and only these ) satisfy: for example, G: = { x | x is an even natural number and greater than 2}, read " Let G be the set of all x for which: x is an even natural number and greater than 2 " or shorter: " Let G be the set of all even natural numbers > 2 ".

It is sometimes difficult to decide whether two sets intensionally described are the same. To this must be determined whether the properties of the intensional descriptions are logically equivalent (if a property is true, there is also the other, and vice versa).

Empty set

The set that contains no elements is called the empty set. It is called with or even and has the power. From the extensionality follows immediately that there is only one empty set: Any " other" empty set, the (so no ) contains the same elements, this would be the same. Consequently, and different, since the latter set contains a quantity other than element.

Subset

A set is part of a set, if every element of is also an element of.

Is then superset ( rare: amount ) called. Formal:

So In particular, any set A subset of itself. The empty set is a subset of every set.

Is a proper subset of ( or is proper subset of it ) when a subset of, but of different, so each element also element of is, but ( at least) one element exists in, which is not included.

The relation " is subset of " is a partial order. The relation " proper subset " is a strict partial order.

There are two notations for subsets in use:

• For " subset " and " proper subset " or
• For " subset " and " proper subset ".

The former corresponds to the system by Bertrand Russell (see Principia Mathematica) introduced and illustrates the analogy to the character and. It is used in this product, but both systems are widely used.

The negation of the relations, and can be described by the crossed each relation symbol, so for example, by. It is also possible to reverse the order of the two arguments as the possibility the relation symbol is inverted. So, instead of also be written instead of also and instead of. A simultaneous strikeout and turning this relation symbols is conceivable.

Intersection (intersection, even " average " )

Given a non-empty set of sets. The intersection ( by intersection) of the amount of the elements included in each element amount. Formal:

Element sets without common elements are called disjoint or disjoint. Their intersection is the empty set.

If a pair of quantity, ie, one writes for

And reads the: cut (or the average of and ) is the set of all elements that are included in both or in.

This notation can be easily generalized to the average of a finite number of quantities.

Different spelling for the average of any number of quantities:

The elements of the set, which themselves are again quantities are denoted by. It is an " index set " (lambda ) introduced so that is. The intersection is then written as:

Thus the set of all elements that are included in all the sets.

An older term for the average is the inner product or product of the first kind, this is then as

Written. In particular, the last notation is considered by many authors for the Cartesian product (see below) reserved and should not be used for the intersection to avoid misunderstandings.

Association ( union )

This is the dual to the intersection of concept: the union of the set of elements that are contained in at least one element of quantity. Formal:

Unlike shall also be declared if it is empty, and indeed arises.

For one writes (similar to the average):

And read this: united with (or: The association of and ) is the set of all elements that are contained in or in. The "or" is not - to be exclusive here. The combination also includes elements that are included in both sets.

If quantities do not contain common elements, so they are disjoint, we also used the sign of the union of these disjoint sets. However, while the sign of the association intuitively with the connective (or ) can be identified, must be distinguished ( or exclusionary ) between the sign of the disjoint union and the connective.

Using a suitable index set to write:

This notation is also suitable for the union of finitely many sets.

As an older term for this, the sum is still sometimes used and then written

Caution: The term sum is now also used for the disjoint union of sets.

Difference and complement

The difference is usually defined only for two quantities: the differential quantity ( also residual amount ) of and the amount of the elements which are, however, not included. Formal:

The difference is in contrast to cutting and union neither commutative nor associative.

If, as is, the difference also complement of in. This term is primarily used when a basic amount, which includes all amounts in question in a particular investigation. This amount must then be hereafter no longer be mentioned, and

Simply means the complement of. Other spellings are, or.

Symmetric difference

The amount

Is called the symmetric difference of and. Is the set of all elements which are each in one, but not in both sets. When using the exclusive-or ( " either - or": or ), you can instead also

. Write

Cartesian product

The product quantity or the Cartesian product is another way of linking quantities. However, the elements of the Cartesian product of two sets are not elements of the output quantities, but more complex objects. Formally, the product quantity is defined by and as

And thus the set of all ordered pairs whose first element and the second element is from. Under the use of n- tuples of the Cartesian product for the link can finally generalize many levels:

If the quantities all the same a lot, so you also writes short for the amount of product. For the amount of product of a family of sets with an arbitrary index set, a general concept of function is needed. It is the set of all functions that assigns each index element is a member of the set, thus

Whether such a Cartesian product is not empty, ie, if there is ever always given such functions as on the right side of the defining equation, is closely connected with the axiom of choice. If the quantities are all equal an amount to write the amount of product as well as short.

Power set

The power set of the set of all subsets of.

The power set of always contains the empty set and the crowd. Thus, so it is a singleton. The power set of a singleton, ie, contains two elements. As a general rule: Do exactly elements, so has the number of elements, that is. This motivates the notation instead.

For infinite sets, the concept is not without problems: There is no proven method that could list all the subsets. (See also:. Cantor's second diagonal argument) In an axiomatic construction of set theory ( ZFC example ) must be called for the existence of the power set by its own power set axiom.

Therefore Constructive mathematicians consider the power set of an infinite set as a basically unfinished area, which - depending on the progress of mathematical research - still new quantities can be added.

Examples of set operations

We consider the quantities and. It shall apply:

• ,
• ,,
• ,,
• ,, ,
• ,,
• = 3, == 2, = 0, = 1
• ,, ,
• ,,
• ,

Concrete examples are named here again.

• The set of all two -digit " numbers liquor " is. 33 is an element of this set 23 is not.
• The set of natural numbers is a proper subset of the set of integers.

Amounts in the logic

Within the science of logic is concerned especially the branch of quantifier ( traditional term predicate logic) with quantities, their elements and the amount of education in the sense of set theory, as well as partially the theory of terms and the logic of science. The investigations are the universal quantifier and the existential quantifier. The universal quantifier is used for example in the sentences. All even numbers are divisible by two without residues; All swans are white. and all metals conduct electricity. The existential quantifier, for example, in the sentences: There are some students who receive a scholarship. and in negated form: There are no black swans. By means of the universal quantifier and existential quantifier as the logical operators and and their negations and logical terms ( eg even number, divisible by two without a remainder ) and quantities are always defined, in the examples, for example, the set of all even numbers, the set of all ( at all existing ) Swans, the set of all metals, the set of all students and of which the subset of students who receive a scholarship.

There is an extensive literature emanating from different quantity terms to the subject matter of this area of logic, ie the definition of the concept of quantity. Abraham Fraenkel and Yehoshua Bar - Hillel described this controversy in 1958 as a modern continuation of the medieval universals. Are predominant in mathematics, the varieties of neo-realism and Platonism as successor to the medieval realism (ideas are real) as they represent particular Alonzo Church, Kurt Gödel and Rudolf Carnap since his work on the logical semantics. This write the amount an independent existence apart from the elements of the set to. Since this leads to contradictions, write some representatives of this direction also the type hierarchy of sets in which they are divided, an independent existence. The nominalist ( Nelson Goodman, Willard Van Orman Quine, Leon Henkin ) compete against this view and assume that quantities exist as abstract objects not specifically next to the items. Subject could only be concrete sensible objects. For nominalists statements about quantities ( and otherwise all abstract objects ) only abbreviated expressions with which we talk about concrete objects. This group also Leon Chwistek and in a more moderate form of Paul Lorenzen heard. The Neoconceptualists including the intuitionists and constructivists consider quantities as constructions. They only accept the intuitively obvious quantities exist ( students who receive a scholarship ) or can be constructed from existing quantities. The Russian logician Alexander Zinoviev proposed to consider the logical concept of class not only as terms, but to define it as it distinguishable additional logical operator to the amount training. To form the class of numbers, it is then sufficient to simply connect the operator to the term. Is the concept known or defined number, the class of numbers produced so with the writing down of the term, or the uttering of " The Class of Numbers". Is God or gods defined (eg in principle immortal, powerful beings ) is formed with the concept class of gods a set whose existence does not depend on whether a single element exists or can exist only. The concept: The class of round squares thus forms a lot, if the properties are known from round and square, even if no round square exists in the universe and can exist in this case.

Further terms

• Subsets of the real line, the plane or the three-dimensional Euclidean space are often called, for historical reasons or to give an indication of the elements contained in point sets. This term testifies to the geometric origin of set theory.
• In modern mathematics, the speed ranges ( with the empty set as the only basic building block ) built purely with the methods of set theory gradually, from the natural numbers, the integers and the rational numbers to the real numbers (and possibly further to the complex numbers and beyond ).
• In school, set theory has at times become very important under the heading New mathematics.
• With endless amounts of special phenomena occur in regard to the usual order relations.
• To illustrate the relationships between quantities are used pie charts.
• Relationships between the elements of a set and which other one are described by "Assignments" (relations), unambiguous assignments through " images" (functions).