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Dalam [[matematika]], '''himpunan''' adalah segala koleksi benda-benda tertentu yang dianggap sebagai satu kesatuan. Walaupun hal ini merupakan [[ide]] yang sederhana, tidak salah jika himpunan merupakan salah satu [[konsep]] penting dan mendasar dalam matematika modern, dan karenanya, studi mengenai struktur kemungkinan himpunan dan [[teori himpunan]], sangatlah berguna.
In mathematics, the set is any collection of certain objects that are considered as a single entity. While this is a simple idea, it is not wrong if the set is one of the important and fundamental concepts in modern mathematics, and therefore, the study of the structure of the set of possibilities and set theory, it is very useful.
[[Berkas:Venn A intersect B.svg|thumb|
 
 
 
Slices of two sets represented by the Venn diagram
 
 
 
 
 
Set theory, newly created in the late 19th century, is now part scattered in mathematics education which was introduced as early as primary school level. This theory is a mathematical language to describe modern. Set theory can be considered as the basis for the construction of nearly all aspects of mathematics and is the source from which all the math derived.
 
 
 
 Notation Association
 
Relations between 8 piece set using a Venn diagram
 
 
 
Typically, the set of names written using capital letters, for example, S, A, or B, while members of the set is lowercase (a, c, z). This way of writing is commonly used, but does not limit that each set should be written that way. The table below shows the format of writing a set of commonly used.
 
 
 
NamaNotasiContoh
 
 
 
 Assemblage
 
 Uppercase
 
 
 
 Member of the set
 
 Lowercase (if it is a letter)
 
 
 
 Classroom
 
 Handwritten letters
 
 
 
The sets of numbers are well known, such as complex numbers, real, round, and so on, using a special notation.
 
 
 
Number
 
 Original
 
 Round
 
Rational
 
 Real
 
Complex
 
 
 
 
 
Notation
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Special symbols used in set theory is:
 
 
 
Symbol
 
Meaning
 
 
 
  or
 
 Empty set
 
 
 
 The combined operation of two sets
 
 
 
 Operation sliced ​​two sets
 
 
 
 ,,,
 
 Subsets, subset true, Superhimpunan, true Superhimpunan
 
 
 
 Complement
 
 
 
 The set of power
 
 
 
 
 
The set can be defined in two ways, namely:
 
 Enumeration, which lists all members of the set. If too much but follow a certain pattern, can use the ellipsis (...).
 
 Builders set, not the register, but by describing the properties that must be met by each member of the set.
 
 
 
Set builder notation can lead to a variety of paradoxes, for example, is the set of the following:
 
A set can not exist, because if A exists, means it must contain a member who is not a member. But if not a member, then how could A could contain the member.
 
 
 
 Empty set
 
Set {apples, oranges, mangoes, bananas} have members of apples, oranges, mangos, and bananas. The other set, such as {5, 6} has two members, namely numbers 5 and 6. We may define a set that does not have any members. This set is called the empty set.
 
 
 
Empty set does not have any members, is written as:
 
 
 
 The relation between the set
 
 
 
 Subsets
 
Of a set, for example, A = {apple, orange, mango, banana}, can be made of other associations whose members are drawn from the set.
 
 {apples, oranges}
 
 {oranges, bananas}
 
 {apple, mango, banana}
 
The third set of the above have a common trait, that every member of the set that are also members of set A. The set-set is referred to as a subset of A. So it can be formulated:
 
 
 
B is a subset of A if every member of B is also present in A.
 
The above sentence is still true for B empty set. Then it is also a subset of A.
 
 
 
For any set A,
 
 
 
The above definition also includes the possibility that a subset of A is A itself.
 
 
 
For any set A,
 
 
 
A subset of the term usually means includes A as a set of its own parts. Sometimes the term is also used to refer to a subset of A, but not A itself. Understanding where used is usually clear from the context.
 
 
 
True subsets of A refers to a subset of A, but does not include a themselves.
 
 
 
 Superhimpunan
 
The opposite of a subset is superhimpunan, ie the set of the larger that includes the set.
 
 
 
 The similarity of two sets
 
Sets A and B are called the same, if every member of A is a member of B, and vice versa, every member of B are members of A.
 
or
 
The above definition is very useful to prove that two sets A and B are the same. First, prove first A is a subset B, then prove that B is a subset of A.
 
 
 
 Counsel Association
 
The set of power or set the power (power set) of A is the set consisting of all subsets of A. notation is.
 
 
 
If A = {apple, orange, mango, banana}, then:
 
 <nowiki>{{</nowiki>},
 
   {apple}, {orange}, {} mango, banana {},
 
   {apples, oranges}, {apple, mango}, {apples, bananas},
 
   {orange, mango}, {oranges, bananas}, {mango, banana},
 
   {apples, oranges, mangoes}, {apples, oranges, bananas}, {apple, mango, banana}, {orange, mango, banana},
 
   {apples, oranges, mangoes, bananas<nowiki>}}</nowiki>
 
 
 
The number of members is contained in the power set of A is 2 to the power many members A.
 
 
 
 Classroom
 
A set is called as a class, or family set if the set is composed of the sets. The set is a set family. Note that for any set A, the set of his power, is a family of sets.
 
 
 
The following examples, not a class, because it contains members c are not set.
 
 
 
 Cardinality
 
Cardinality of a set can be understood as a measure of the number of members contained by the set. The number of members of the set is 4. The set also has a number of members of the two sets 4. Means equivalent to each other, or is said to have the same cardinality.
 
 
 
Two sets A and B have the same cardinality, if there is one-to-one function that maps A to B. Because it is easy to create a function that maps one-on-one and to set A to B, then the two sets have the same cardinality ,
 
 
 
 The set Denumerabel
 
If a set is equivalent to the set, ie the set of natural numbers, then the set is called denumerabel. Cardinality of the set is referred to as cardinality.
 
 
 
The set of all positive even number is set denumerabel, because it has one-one correspondence between the set is the set of natural numbers, which is expressed by.
 
 
 
 Finite set
 
If a set has cardinality less than the cardinality, then the set is a finite set.
 
 
 
 The set Tercacah
 
The set is called tercacah if the set is finite or denumerabel.
 
 
 
 Association of Non-Denumerabel
 
Tercacah set that is not called the set of non-denumerabel. Examples of this set is the set of all real numbers. The cardinality of the set of this type is referred to as cardinality. Proving that the real numbers are not denumerabel can use diagonal proof.
 
 
 
The set of real numbers in the interval (0,1) also has cardinality, because there is a one-one correspondence of the set with the set of all real numbers, one of which is.
 
 
 
 Characteristic Functions
 
Function characteristics indicate whether a member contained in a set or not.
 
If so:
 
There is a one-one correspondence between the set of power by the set of all functions characteristic of S. This results we can write the set as a sequence of numbers 0 and 1, which states whether there is a member in the set.
 
 
 
 Binary representation
 
If the context of the conversation is on a set universe of S, then every subset of S can be written in rows of numbers 0 and 1, also called binary form. Binary numbers using the numbers 1 and 0 in each digit. Each bit position associated with each member of S, so that the value of 1 indicates that the members there, and a value of 0 indicates that the member is not there. In other words, each bit is the characteristic function of the set.
 
For example, if the set S = {a, b, c, d, e, f, g}, A = {a, c, e, f} and B = {b, c, d, f}, then:
 
 
 
 Binary Representation Association
 
 ---------------------------- -------------------
 
                                     a b c d e f g
 
 S = {a, b, c, d, e, f, g} -> 1 1 1 1 1 1 1
 
 A = {a, c, e, f} -> 1 0 1 0 1 1 0
 
 B = {b, c, d, f} -> 0 1 1 1 0 1 0
 
 
 
How to denote the set like this is very advantageous to perform set operations, such as union (combined), intersection (sliced), and complement (complement), since we live using bit operations to do so. Representation set in binary form used by the compiler-compiler Pascal and Delphi.
 
 
 
 The basic operation
 
 Combined
 
Combined between sets A and B.
 
Two sets or more are combined together. The combined operation <nowiki>{{</nowiki>nowrap | 1 = A ∪ B is equivalent to A or B, and himpunannya members are all members of which included the set of A or B.
 
 
 
Examples:
 
 {1, 2} ∪ {1, 2} = {1, 2}.
 
 {1, 2} ∪ {2, 3} = {1, 2, 3}.
 
 Budi {} ∪ {} = {Budi Dani, Dani}.
 
 
 
Some basic properties combined:
 
 A ∪ B = B ∪ A.
 
 A ∪ (B ∪ C) = (A ∪ B) ∪ C.
 
 A ⊆ (A ∪ B).
 
 A ∪ A = A.
 
 A ∪ ∅ = A.
 
 A ⊆ B if and only if A ∪ B = B.
 
 
 
 Sliced
 
Wedge between sets A and B.
 
A ∩ B slices operation equivalent to A and B. Sliced ​​a new set whose members consist of members that are shared between two or more sets are connected. If A ∩ B = ∅, then A and B can be said to be disjoint (separate).
 
 
 
Examples:
 
 {1, 2} ∩ {1, 2} = {1, 2}.
 
 {1, 2} ∩ {2, 3} = {2}.
 
 {Budi, Cici} ∩ {Dani, Cici Cici} = {}.
 
 {Budi} ∩ {Dani} = ∅.
 
 
 
Some basic properties of slices:
 
 A ∩ B = B ∩ A.
 
 A ∩ (B ∩ C) = (A ∩ B) ∩ C.
 
 A ∩ B ⊆ A.
 
 A ∩ A = A.
 
 A ∩ ∅ = ∅.
 
 A ⊂ B if and only if A ∩ B = A.
 
 
 
 Complement
 
Complement B to A.
 
A complement to the U.
 
Diferensi symmetric sets A and B.
 
A complementary operation ^ C is equivalent to not A or A '. Complement operation is an operation whose members consist of members outside the set.
 
 
 
Examples:
 
 {1, 2} \ {1, 2} = ∅.
 
 {1, 2, 3, 4} \ {1, 3} = {2, 4}.
 
 
 
Some basic properties of complements:
 
 A \ B ≠ B \ A for A ≠ B.
 
 A ∪ A '= U.
 
 A ∩ A '= ∅.
 
 (A ')' = A.
 
 A \ A = ∅.
 
 U '= ∅ and ∅' = U.
 
 A \ B = A ∩ B '.
 
 
 
Extension of the complement is diferensi symmetric (reducing the set), when applied to sets A and B or A - B produce
 
 
 
 
 
For example, diferensi symmetrical between:
 
 {7,8,9,10} and {9,10,11,12} is {7,8,11,12}.
 
 {Ana, Budi, Smith, Felix} and {Cici, Budi, Smith, Ela} is {Ana, Cici, Ela, Felix}.
 
 
 
 Kali results Cartesian
 
Kertesian product (multiplication set) AXB (A and B) and a member of the set A = {x, y, z} and B = {1,2,3}.
 
Kali Cartesian or multiplication result is a set of operations that combine members of a set with another set. Multiplication set between A and B is defined by A × B. Members set | A × B | is the ordered pair (a, b) where a is a member of the set A and B is a member of the set B.
 
 
 
Examples:
 
 {1, 2} × {x, y} = {(1, x), (1, y), (2, x), (2, y)}.
 
 {1, 2} × {a, b, c} = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c) }.
 
 {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.
 
 
 
Some basic properties of the set of multiplication:
 
 A × ∅ = ∅.
 
 A × (B ∪ C) = (A × B) ∪ (A × C).
 
 (A ∪ B) × C = (A × C) ∪ (B × C).
 
 | A × B | = | A × B | = | A | × | B |.
 
 
 
 
 
 
 
 
 
Wikibooks has a book titled
 
Materials: Association
 
 
 
 Reference
 
 Lipschutz, S. Set Theory. McGraw-Hill
 
 Delphi 5 Memory Management
 
 
 
 Further reading
 
 
 
 
 
 
 
Wikimedia Commons has media related to: Association (math)
 
<nowiki>Dauben, Joseph W., Georg Cantor: His Mathematics and Philosophy of the Infinite, Boston: Harvard University Press (1979) ISBN 978-0-691-02447-9.</nowiki>
 
<nowiki> Halmos, Paul R., Naive Set Theory, Princeton, NJ: Van Nostrand (1960) ISBN 0-387-90092-6</nowiki>
 
<nowiki> Stoll, Robert R., Set Theory and Logic, Mineola, NY: Dover Publications (1979) ISBN 0-486-63829-4</nowiki>
 
 Velleman, Daniel, How To Prove It: A Structured Approach, Cambridge University Press (2006) ISBN 978-0-521-67599-4Dalam [[matematika]], '''himpunan''' adalah segala koleksi benda-benda tertentu yang dianggap sebagai satu kesatuan. Walaupun hal ini merupakan [[ide]] yang sederhana, tidak salah jika himpunan merupakan salah satu [[konsep]] penting dan mendasar dalam matematika modern, dan karenanya, studi mengenai struktur kemungkinan himpunan dan [[teori himpunan]], sangatlah berguna.[[Berkas:Venn A intersect B.svg|thumb|
Irisan dari dua himpunan yang dinyatakan dengan [[diagram Venn]]
]]