Empty set

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In mathematics, and more specifically set theory, the empty set or null set is a unique set that has no elements; size or cardinality (element count in a set) is zero. Some axiomatic set theory ensures that empty sets exist by entering the axioms of empty sets; in another theory, its existence can be inferred. Many set properties that allow completely empty for empty sets.

Null set is also used as a technical term in measure theory. In that context, he describes a set of zero sizes; Such sets are not necessarily empty. An empty set may also be called the void set .

Video Empty set

Notation

Common notation for empty set includes "{}", " ${\ displaystyle \ emptyset}$ ", and"? "The last two symbols introduced by the Bourbaki group (especially Andrà © Weil) in 1939, inspired by the letters ÃÆ'ËÅ" in the Norwegian and Danish Alphabet (and are not related in any way with the Greek letters?). In the past," 0 " sometimes used as a symbol for empty sets, but these are now considered as improper use of notation.

Symbols? available in Unicode point U 2205. It can be encoded in HTML as & amp; emptyset; and as & amp; # 8709; . This can be encoded in LaTeX as \ varnothing . Symbols ${\ displaystyle \ emptyset}$ encoded in LaTeX as \ emptyset ; it's not available in HTML/Unicode.

Maps Empty set

Properties

In standard axiomatic set theory, with the principle of extensionality, two sets are equal if they have the same element; therefore there is only one set without elements. Therefore there is only one empty set, and we are talking about "empty sets" rather than "empty sets".

The mathematical symbols used below are described here.

Untuk set apa saja A :

• Set kosong adalah bagian dari A :

  ${\ displaystyle \ forall A: \ varnothing \ subseteq A}  ÂÂ$

• Penggabungan A dengan set kosong adalah A :

  ${\ displaystyle \ forall A: A \ cup \ varnothing = A}  ÂÂ$

• Persimpangan A dengan set kosong adalah set kosong:

  ${\ displaystyle \ forall A: A \ cap \ varnothing = \ varnothing}  ÂÂ$

• Produk Cartesian dari A dan himpunan kosong adalah himpunan kosong:

  ${\ displaystyle \ forall A: A \ times \ varnothing = \ varnothing}  ÂÂ$

Set kosong memiliki properti berikut:

• Satu-satunya bagiannya adalah set kosong itu sendiri:

  ${\ displaystyle \ forall A: A \ subseteq \ varnothing \ Rightarrow A = \ varnothing}  ÂÂ$

• Power set dari set kosong adalah himpunan yang hanya berisi set kosong:

  ${\ displaystyle 2 ^ {\ varnothing} = \ {\ varnothing \}}  ÂÂ$

• Jumlah elemennya (yaitu, kardinalitasnya) nol:

  ${\ displaystyle \ mathrm {card} (\ varnothing) = 0}  ÂÂ$

The connection between the empty and zero sets goes further, however: in the set-theoretic definition of the natural number standard, the set is used to model the original number. In this context, zero is modeled by an empty set.

Untuk properti apa pun:

• Untuk setiap elemen ${\ displaystyle \ varnothing}  ÂÂ$ properti memegang (kebenaran hampa);
• Tidak ada elemen ${\ displaystyle \ varnothing}  ÂÂ$ yang properti tersebut tahan.

Conversely, if for some properties and some set V , the following two statements apply:

• For each element V property is retained;
• There is no V element that the property belongs to,

then ${\ displaystyle V = \ varnothing}$ .

Dengan definisi subset, himpunan kosong adalah himpunan bagian dari himpunan apa pun A . Yaitu, setiap elemen x dari ${\ displaystyle \ varnothing}  ÂÂ$ milik A . Memang, jika tidak benar bahwa setiap elemen ${\ displaystyle \ varnothing}  ÂÂ$ ada di A maka akan ada setidaknya satu elemen ${\ displaystyle \ varnothing}  ÂÂ$ yang tidak ada di A . Karena ada no elemen ${\ displaystyle \ varnothing}  ÂÂ$ sama sekali, tidak ada elemen ${\ displaystyle \ varnothing}  ÂÂ$ yang tidak ada di A . Pernyataan apa pun yang dimulai "untuk setiap elemen ${\ displaystyle \ varnothing}  ÂÂ$ "tidak membuat klaim substantif; itu adalah kebenaran hampa. Ini sering diparafrasekan sebagai" semuanya benar dari elemen-elemen set kosong. "

Operasi pada set kosong

When talking about the number of elements from a finite set, one must lead to the convention that the number of elements of an empty set is zero. The reason is that zero is the element of identity for addition. Similarly, the product of the empty set elements should be considered one (see empty product), because that one is the identity element for multiplication.

Disturbance is a permutation of a set with no fixed point. Empty sets can be considered as chaos itself, because it has only one permutation ( ${\ displaystyle 0! = 1}$ ), and it's true that no element (of the empty set) can be found that retains its original position.

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In other math areas

Expanded real number

Since the empty set has no members, when it is considered part of a sorted set, each member of that set will be the upper and lower bounds for the empty set. For example, when considered as a subset of real numbers, in the usual order, represented by a real number line, each real number is an upper and lower bound for an empty set. When considered as part of an extended extension formed by adding two "numbers" or "points" to a real number, ie infinitely negative, denoted ${\ displaystyle - \ infty \! \ ,,}$ defined less than any other widespread real number, and positive infinity, denoted ${\ displaystyle \ infty \! \ ,,}$ defined more than any other real number, then:

${\ displaystyle \ sup \ varnothing = \ min (\ {- \ infty, \ infest \} \ cup \ mathbb {R}) = - \ infty, }$

dan

${\ displaystyle \ inf \ varnothing = \ max (\ {- \ infty, \ infty \} \ cup \ mathbb {R}) = \ infty.}  ÂÂ$

That is, the lower upper limit (sup, or supremum) of the empty set is a negative infinity, while the largest lower bound (inf or infimum) is positive infinity. By analogy with the above, in the domain of the real extension, negative infinity is the identity element for the maximum operator and supremum, while positive infinity is the element of identity for the minimum and infimum.

Topology

In any topology space X , the empty set is open by definition, such as X . Because the complement of the open set is closed and the set is empty and X complement each other, the empty set is also closed, making it a set of clopen. What's more, the empty set is compact by the fact that every limited set is compact.

Closing empty empty set. This is known as "the preservation of battal unity".

Category theory

If A is a set, then there is a function f of {} to A , the function is empty. As a result, the empty set is a unique start object of the set and function categories.

Empty sets can be converted into topology space, called empty space, in only one way: by defining empty sets to open. This empty topology space is a unique initial object in the topological space category with a continuous map. In fact, this is a strict initial object: only an empty set that has a function to the empty set.

Set theory

Dalam konstruksi von Neumann dari ordinal, 0 didefinisikan sebagai set kosong, dan penerus ordinal didefinisikan sebagai ${\ displaystyle S (\ alpha) = \ alpha \ cup \ {\ alpha \}}  ÂÂ$ . Dengan demikian, kami memiliki ${\ displaystyle 0 = \ varnothing}  ÂÂ$ , ${\ displaystyle 1 = 0 \ cup \ {0 \} = \ {\ varnothing \}}  ÂÂ$ , ${\ displaystyle 2 = 1 \ cup \ {1 \} = \ {\ {\ varnothing \}, \ varnothing \}}  ÂÂ$ , dan seterusnya. Konstruksi von Neumann, bersama dengan aksioma ketidakterbatasan, yang menjamin keberadaan setidaknya satu set tak terbatas, dapat digunakan untuk membangun himpunan bilangan natural, ${\ displaystyle \ mathbb {N} _ {0}}  ÂÂ$ , sedemikian sehingga aksioma Peano dari aritmatika dipenuhi.

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Keberadaan yang dipertanyakan

Teori set aksiomatik

In the Zermelo set theory, the existence of an empty set is ensured by the axiom of the empty set, and its uniqueness follows the axiom of extensionality. However, the empty set axioms can be displayed redundantly in one of two ways:

• There is already an axiom that implies at least one set. Given such axioms along with the axiom of separation, the existence of empty sets is easily proven.
• In the presence of urelements, it is easy to prove that at least one set exists, ie. the set of all urelement (assuming no exact class of them). Again, given the axiom of separation, the empty set is easily proven.

Philosophical issues

While the empty set is a standard and widely accepted mathematical concept, it remains an ontological curiosity, whose meaning and usefulness is disputed by philosophers and logicians.

Empty set is not equal to nothing ; more precisely, it is a set of nothing inside and a set is always something . This problem can be solved by seeing a set as empty bags no doubt is still there. Darling (2004) explains that the empty set is nothing but the "set of all triangles with four sides, a set of all numbers greater than nine but smaller than eight, and a set of all opening movements in chess involving a king."

Popular silogism

There is nothing better than eternal bliss; ham sandwich is better than nothing at all; therefore, ham sandwiches are better than lasting bliss

often used to denote the philosophical relationship between any concept and empty set. Darling writes that contrast can be seen by rewriting the statement "Nothing better than lasting happiness" and "Sandwich ham A is better than nothing" in mathematical tone. According to Darling, the first is equivalent to "The set of all things better than eternal happiness is ${\ displaystyle \ varnothing}$ "and the last to" Groups {ham sandwich} is better than group ${\ displaystyle \ varnothing}$ "It should be noted that the first compares the set element, while the second compares the set itself.

Jonathan Lowe argues that while the set is empty:

"... is undoubtedly an important milestone in the history of mathematics,... we should not assume that its utility in the calculations depends on what actually points to some object."

it also happens that:

"Everything we ever knew about the empty set is that it (1) is a set, (2) has no members, and (3) is unique among sets in not having members. many things are 'non-members', in a set-theoretical sense - that is, all non-sets.It is very clear why these things do not have members, because they are not set.What is unclear is how it can be, set, a set that has no members We can not juggle such entity into existence with any provision. "

George Boolos argues that much of what up until now is obtained by set theory can be easily obtained by quantifying plural over individuals, without re-establishing it as a single entity that has another entity as a member.

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See also

• Set is populated
• Nothing

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References

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Further reading

• Halmos, Paul, Naive Set Theory . Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBNÃ, 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. ISBNÃ, 978-1-61427-131-4 (Paperback edition).
• Jech, Thomas (2002), Set Theory , Springer Monographs in Mathematics (3rd Millennium Edition), Springer, ISBN 3- 540 -44085-2
• Graham, Malcolm (1975), Modern Basic Mathematics (Hardcover) (2nd ed.), New York: Harcourt Brace Jovanovich, ISBN: 0155610392

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External links

• Weisstein, Eric W. "Empty Set". MathWorld .

Source of the article : Wikipedia