离散数学与集合论

集合论 (Set theory)

The set is a well-defined collection of definite objects of perception or thought and the 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.

集合是定义明确的感知或思想客体的集合，而 ( 是集合论之父。 集合也可以被认为是将单个对象组合成一个整体。 这些对象应彼此不同，并且应与所有未与正在考虑的对象集中的对象区分开。 因此，st可以是一串葡萄，茶具，也可以由几何点或直线组成。

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.

集合定义为相同类型的不同元素的无序集合，其中类型由集合的编写者定义。

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，集合用大写字母表示，集合的母版或元素用{括起来。

1 E A →   1 belong to A    1 E/ A  → 1 does not belong to A

套装类型 (Types of set)

There are many types of set in the set theory:

集合论中有许多类型的集合：

1. Singleton set

1.单身套装

If a set contains only one element it is called to be a singleton set.

如果一个集合仅包含一个元素，则称其为单例集合。

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}给出的集合都仅包含一个元素，因此是单例集合。

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

由自然数的对象组成的集合，即其中数字元素是有限的，被称为有限集合。 考虑集合

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.

显然， A ， B包含有限数量的元素，即A中的 4个对象和B中的 6个对象。 因此，它们是有限集。

3. Infinite set

3.无限集

If the number of elements in a set is finite, the set is said to be an infinite set.

如果集合中元素的数量是有限的，则将该集合称为无限集合。

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，所有自然数的集合由N = {1，2，3，...}给出，是一个无限集合。 类似地，)和1之间的所有有理数的集合由

A = {x:x E Q, 0 <x<1} is an infinite set.

A = {x：x EQ，0 <x <1}是一个无限集。

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.

由相同元素组成的两组A和B被称为相等组。 换句话说，如果集合A中的一个元素集合，则集合A和B称为相等，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.

空集或空集是没有成员的有效集。

A = { } / phie cardinality of A is 0.

A = {} / A的phie基数为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.

有两种流行的表示形式，即空花括号{}或特殊符号phie 。 这个A是一组具有里面是空集。

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.

一个子集被认为是如果每属于A类元素也属于B B的子集。

A = { 1, 2, 3}    B = { 1, 2, 3, 4}    A subset of B.

7. Proper set

7.正确设置

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.

一组被认为是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包含至少一个元件，其不属于甲 。

8. Improper set

8.设置不当

Set A is called an improper subset of B if and Only if A = B. Every set is an improper subset of itself.

当且仅当A = B时，集合A称为B的不正确子集。 每个集合都是其自身的不适当子集。

9. Power set

9.功率设定

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.

一组的幂集定义为每个可能子集的一组。 如果A的基数为n ，则幂集的基数为2 ^ n，因为每个元素都有两个选项或不属于一个子集。

10. Universal set

10.通用套装

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.

任何正在考虑的所有集合的超集的集合都称为通用集合，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.

翻译自:

离散数学与集合论