離散數學與集合論
集合論 (Set theory)
The set is a well-defined collection of definite objects of perception or thought and the Georg Cantor is the father of set theory. A set may also be thought of as grouping together of single objects into a whole. The objects should be distinct from each other and they should be distinguished from all those objects that do not from the set under consideration. Hence an st may be a bunch of grapes, a tea set or it may consist of geometrical points or straight lines.
集合是定義明確的感知或思想客體的集合，而喬治·康托 ( Georg Cantor)是集合論之父。 集合也可以被認為是將單個對象組合成一個整體。 這些對象應彼此不同，并且應與所有未與正在考慮的對象集中的對象區分開。 因此，st可以是一串葡萄，茶具，也可以由幾何點或直線組成。
A set is defined as an unordered collection of distinct elements of the same type where type is defined by the writer of the set.
集合定義為相同類型的不同元素的無序集合，其中類型由集合的編寫者定義。
Generally, a set is denoted by a capital symbol and the master or elements of a set are separated by an enclosed in { }.
通常，集合用大寫字母表示，集合的母版或元素用{括起來。
1 E A → 1 belong to A
1 E/ A → 1 does not belong to A
套裝類型 (Types of set)
There are many types of set in the set theory:
集合論中有許多類型的集合：
1. Singleton set
1.單身套裝
If a set contains only one element it is called to be a singleton set.
如果一個集合僅包含一個元素，則稱其為單例集合。
Hence the set given by {1}, {0}, {a} are all consisting of only one element and therefore are singleton sets.
因此， {1}，{0}，{a}給出的集合都僅包含一個元素，因此是單例集合。
2. Finite Set
2.有限集
A set consisting of a natural number of objects, i.e. in which number element is finite is said to be a finite set. Consider the sets
由自然數的對象組成的集合，即其中數字元素是有限的，被稱為有限集合。 考慮集合
A = { 5, 7, 9, 11} and B = { 4 , 8 , 16, 32, 64, 128}
A = {5，7，9，11}和B = {4，8，16，32，64，128}
Obviously, A, B contain a finite number of elements, i.e. 4 objects in A and 6 in B. Thus they are finite sets.
顯然， A ， B包含有限數量的元素，即A中的 4個對象和B中的 6個對象。 因此，它們是有限集。
3. Infinite set
3.無限集
If the number of elements in a set is finite, the set is said to be an infinite set.
如果集合中元素的數量是有限的，則將該集合稱為無限集合。
Thus the set of all natural number is given by N = { 1, 2, 3, ...} is an infinite set. Similarly the set of all rational number between ) and 1 given by
因此，所有自然數的集合由N = {1，2，3，...}給出，是一個無限集合。 類似地，)和1之間的所有有理數的集合由
A = {x:x E Q, 0 <x<1} is an infinite set.
A = {x：x EQ，0 <x <1}是一個無限集。
4. Equal set
4.等分
Two set A and B consisting of the same elements are said to be equal sets. In other words, if an element of the set A sets the set A and B are called equal i.e. A = B.
由相同元素組成的兩組A和B被稱為相等組。 換句話說，如果集合A中的一個元素集合，則集合A和B稱為相等，即A = B。
5. Null set/ empty set
5.空集/空集
A null set or an empty set is a valid set with no member.
空集或空集是沒有成員的有效集。
A = { } / phie cardinality of A is 0.
A = {} / A的phie基數為0。
There is two popular representation either empty curly braces { } or a special symbol phie. This A is a set which has null set inside it.
有兩種流行的表示形式，即空花括號{}或特殊符號phie 。 這個A是一組具有里面是空集。
6. Subset
6.子集
A subset A is said to be subset of B if every elements which belongs to A also belongs to B.
一個子集被認為是如果每屬于A類元素也屬于B B的子集。
A = { 1, 2, 3}
B = { 1, 2, 3, 4}
A subset of B.
7. Proper set
7.正確設置
A set is said to be a proper subset of B if A is a subset of B, A is not equal to B or A is a subset of B but B contains at least one element which does not belong to A.
一組被認為是B的真子集，如果A是B的子集，A不等于B或A是B的子集，但B包含至少一個元件，其不屬于甲 。
8. Improper set
8.設置不當
Set A is called an improper subset of B if and Only if A = B. Every set is an improper subset of itself.
當且僅當A = B時，集合A稱為B的不正確子集。 每個集合都是其自身的不適當子集。
9. Power set
9.功率設定
Power set of a set is defined as a set of every possible subset. If the cardinality of A is n than Cardinality of power set is 2^n as every element has two options either to belong to a subset or not.
一組的冪集定義為每個可能子集的一組。 如果A的基數為n ，則冪集的基數為2 ^ n，因為每個元素都有兩個選項或不屬于一個子集。
10. Universal set
10.通用套裝
Any set which is a superset of all the sets under consideration is said to be universal set and is either denoted by omega or S or U.
任何正在考慮的所有集合的超集的集合都稱為通用集合，并用omega或S或U表示 。
Let A = {1, 2, 3}
C = { 0, 1} then we can take
S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} as universal set.
翻譯自: https://www.includehelp.com/basics/set-theory-and-types-of-set-in-discrete-mathematics.aspx
離散數學與集合論
