離散數學和組合數學什么關系

關系類型 (Types of Relation)

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

集合之間存在許多類型的關系，

1. Universal Relation

1.普遍關系

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

從組a到b關系R被認為是通用的，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)是普遍關系。

2. Compliment Relation

2.稱贊關系

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

關系的稱贊將包含所有對，其中對不屬于關系而是屬于笛卡爾積。

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.空關系

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是空關系

4. Inverse of relation

4.關系逆

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:

關系的逆由R ^ -1表示， R ^ -1是只是以不同或相反順序寫入的同一對對的集合。 令R為從A到B的任何關系。 R的逆表示由R ^ -1是從B到A的關系由下式定義：

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

5. Composite Relation

5.復合關系

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為任意三個集合。 讓我們考慮從A到B的關系R和從B到C的另一個關系。 兩個關系R和S的組成關系是從集合A到集合C的一個關系，用RoS表示，并定義如下：

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，當EA，c EC時具有(a，b)ER和(b，c)E s
因此，(a，b)ER(b，c)ES =>(a，c)E RoS 。

6. Equivalence Relation

6.等價關系

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.

關系R如果滿足集合x中的自反，對稱和可傳遞的三個屬性，則稱為等價關系。 如果R是集合X中的等價關系，則D(R)的R域是X本身。 因此， R將被稱為X上的關系。

The following are some examples of the equivalence relation:

以下是等價關系的一些示例：

• Equality of numbers on a set of real numbers.

  一組實數上的數字相等。

• 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.

  線的關系是平面中線的平行起點。

• 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鎮中居住的關系。

7. Partial order relation

7.偏序關系

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

假設R是集合A中的一個關系，那么，如果R被稱為偏序關系，

• R is reflexive

  R是反身的

  i.e. aRa ,a belongs to A

  即aRa，a屬于A

• R is anti- symmetric

  R是反對稱的

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

  即aRb，bRa => a = b，a，b屬于a

• R is transitive

  R是可傳遞的

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

  aRb，bRc => aRc，a，b，c屬于A

8. Antisymmetric Relation

8.反對稱關系

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

如果對于x，y，則對集合a的關系R稱為反對稱關系，對于x，y =>

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

如果(x，y)和(y，x)ER，則x = y

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

示例：{(1，2)(2，3)，(2，2)}是反對稱關系。

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

反對稱關系與非對稱關系不相同。 一個關系可以同時是反對稱的和對稱的。

9. Irreflective relation

9.反射關系

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

關系R被說成是對irreflective關系如果x E中的(X，X)不屬于R上 。

Example:

例：

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

10. Not Reflective relation

10.非反思關系

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，則關系R被認為是不反射的。

翻譯自: https://www.includehelp.com/basics/types-of-relation-discrete mathematics.aspx

離散數學和組合數學什么關系