离散数学和组合数学什么关系

关系类型 (Types of Relation)

There are many types of relation which is exist between the sets,

集合之间存在许多类型的关系，

1. Universal Relation

1.普遍关系

A relation r from set a to B is said to be universal if: R = A * B

从组a到b关系R被认为是通用的，如果：R = A * B

Example:

例：

A = {1,2} B = {a, b}

A = {1,2} B = {a，b}

R = { (1, a), (1, b), (2, a), (2, b) is a universal relation.

R = {(1，a)，(1，b)，(2，a)，(2，b)是普遍关系。

2. Compliment Relation

2.称赞关系

Compliment of a relation will contain all the pairs where pair do not belong to relation but belongs to Cartesian product.

关系的称赞将包含所有对，其中对不属于关系而是属于笛卡尔积。

R = A * B – X

R = A * B – X

Example:

例：

A = { 1, 2}   B = { 3, 4}
R = { (1, 3) (2, 4) }
Then the complement of R
Rc = { (1, 4) (2, 3) }

3. Empty Relation

3.空关系

A null set phie is subset of A * B.

空集phie是A * B的子集。

R = phie is empty relation

R = phi是空关系

4. Inverse of relation

4.关系逆

An inverse of a relation is denoted by R^-1 which is the same set of pairs just written in different or reverse order. Let R be any relation from A to B. The inverse of R denoted by R^-1 is the relation from B to A defined by:

关系的逆由R ^ -1表示， R ^ -1是只是以不同或相反顺序写入的同一对对的集合。 令R为从A到B的任何关系。 R的逆表示由R ^ -1是从B到A的关系由下式定义：

 R^-1 = { (y, x) : yEB, xEA, (x, y) E R}

5. Composite Relation

5.复合关系

Let A, B, and C be any three sets. Let consider a relation R from A to B and another relation from B to C. The composition relation of the two relation R and S be a Relation from the set A to the set C, and is denoted by RoS and is defined as follows:

令A ， B和C为任意三个集合。 让我们考虑从A到B的关系R和从B到C的另一个关系。 两个关系R和S的组成关系是从集合A到集合C的一个关系，用RoS表示，并定义如下：

Ros = { (a, c) : an element of B such that (a, b) E R and (b, c) E s, when a E A , c E C}
Hence, (a, b) E R (b, c) E S => (a, c) E RoS.

Ros = {(a，c)：B的元素，当EA，c EC时具有(a，b)ER和(b，c)E s
因此，(a，b)ER(b，c)ES =>(a，c)E RoS 。

6. Equivalence Relation

6.等价关系

The relation R is called equivalence relation when it satisfies three properties if it is reflexive, symmetric, and transitive in a set x. If R is an equivalence relation in a set X then D(R) the domain of R is X itself. Therefore, R will be called a relation on X.

关系R如果满足集合x中的自反，对称和可传递的三个属性，则称为等价关系。 如果R是集合X中的等价关系，则D(R)的R域是X本身。 因此， R将被称为X上的关系。

The following are some examples of the equivalence relation:

以下是等价关系的一些示例：

• Equality of numbers on a set of real numbers.

  一组实数上的数字相等。

• Equality of subsets of a universal set.

  通用集的子集的相等性。

• Similarities of triangles on the set of triangles.

  三角形集上三角形的相似性。

• Relation of lines being a parallel onset of lines in a plane.

  线的关系是平面中线的平行起点。

• Relation of living in the same town on the set of persons living in Canada.

  在加拿大居住的同一套城镇中居住的关系。

7. Partial order relation

7.偏序关系

Let, R be a relation in a set A then, R is called partial order Relation if,

假设R是集合A中的一个关系，那么，如果R被称为偏序关系，

• R is reflexive

  R是反身的

  i.e. aRa ,a belongs to A

  即aRa，a属于A

• R is anti- symmetric

  R是反对称的

  i.e. aRb, bRa => a = b, a, b belongs to a

  即aRb，bRa => a = b，a，b属于a

• R is transitive

  R是可传递的

  aRb, bRc => aRc, a, b, c belongs to A

  aRb，bRc => aRc，a，b，c属于A

8. Antisymmetric Relation

8.反对称关系

A relation R on a set a is called on antisymmetric relation if for x, y if for x, y =>

如果对于x，y，则对集合a的关系R称为反对称关系，对于x，y =>

If (x, y) and (y, x) E R then x = y

如果(x，y)和(y，x)ER，则x = y

Example: { (1, 2) (2, 3), (2, 2) } is antisymmetric relation.

示例：{(1，2)(2，3)，(2，2)}是反对称关系。

A relation that is antisymmetric is not the same as not symmetric. A relation can be antisymmetric and symmetric at the same time.

反对称关系与非对称关系不相同。 一个关系可以同时是反对称的和对称的。

9. Irreflective relation

9.反射关系

A relation R is said to be on irreflective relation if x E a (x ,x) does not belong to R.

关系R被说成是对irreflective关系如果x E中的(X，X)不属于R上 。

Example:

例：

    a = {1, 2, 3}
R = { (1, 2), (1, 3) if is an irreflexive relation

10. Not Reflective relation

10.非反思关系

A relation R is said to be not reflective if neither R is reflexive nor irreflexive.

如果R既不是自反的也不是自反的，则关系R被认为是不反射的。

翻译自: https://www.includehelp.com/basics/types-of-relation-discrete mathematics.aspx

离散数学和组合数学什么关系