离散数学关系的性质_关系和关系的性质| 离散数学

离散数学关系的性质

笛卡尔积(A * B不等于B * A) (Cartesian product (AB not equal to BA))

Cartesian product denoted by * is a binary operator which is usually applied between sets. It is a set of ordered pairs where the first member of the pair belongs to the first set and the second member of the pair belongs second sets.

用*表示的笛卡尔积是二进制运算符，通常应用于集合之间。它是一组有序对，其中该对的第一成员属于第一集合，而该对的第二成员属于第二集合。


If,
|A|     = m |B| = n
|A*B|   = mn

Example:

例：

    A = {1,2}  B = {a, b, c}
A * B = { (1,a), (1,b), (1,c), (2,a), (2,b), (2,c)}

关系 (Relation)

The word relation suggests some familiar example relations such as the relation of father to son, mother to son, brother to sister etc. Familiar examples in arithmetic are relation such as "greater than", "less than", or that of equality between the two real numbers.

关系一词表示一些熟悉的示例关系，例如父亲与儿子的关系，母亲与儿子的关系，兄弟与姐妹的关系等。算术中的常见示例是诸如“大于” ， “小于”或关系之间相等的关系。两个实数。

Here, we shall only consider relation called binary relation, between the pairs of objects. Before we give a set-theoretic definition of a relation we note that a relation between two objects can be defined by listing the two objects an ordered pair.

在这里，我们将仅考虑对象对之间的称为二进制关系的关系。在给出关系的集合理论定义之前，我们注意到可以通过将两个对象按有序对列出来定义两个对象之间的关系。

Definition:

定义：

Any set of ordered pairs defines a binary relations. We shall call a binary relation simply a relation. It is sometimes convenient to express the fact that particular ordered pair say (x,y) E R where, R is a relation by writing xRY which may be read as "x is a relation R to y".

任何一组有序对都定义了二进制关系。我们将二元关系简称为关系。通过写xRY可以表达特定的有序对说(x，y)ER的事实，其中R是一个关系，可以读为“ x是R与y的关系” 。

Example:

例：

The relation of father to his child can be described by a set , say ordered pairs in which the first member is the name of the father and second the name of his child that is:

父亲与他的孩子的关系可以用集合(例如有序对)描述，其中第一个成员是父亲的名字，第二个成员是他的孩子的名字，即：

F = { (x , y) |x is the father of y}

F = {(x，y)| x是y的父亲}

域 (Domain)

Let, S be a binary relation. The set D(S) of all objects x such that for some y, (x,y) E S is said to be the domain of S.

设S为二元关系。所有对象x的集合D(S)使得对于某个y ， (x，y)ES被称为S的域。

范围 (Range)

The set R(S) of all objects y such that for some x, (x,y) E S said to be the range of S.

所有对象y的集合R(S) ，使得对于某些x ， (x，y)ES表示为S的范围。

Let r A B be a relation

令r AB为关系

    DOM(R) = {a|(a, b)E R for some b E B} 
Range(R) = {b |(a, b) E R } for some

Properties of binary relation in a set

集合中二进制关系的属性

There are some properties of the binary relation:

二进制关系具有一些属性：

A binary relation R is in set X is reflexive if , for every x E X , xRx, that is (x, x) E R or R is reflexive in X <==> (x) (x E X -> xRX).
如果对于每个x EX ， xRx ，即(x，x)ER或R在X <==>(x)(x EX-> xRX)中是自反的，则集合X中的二元关系R是自反的。

The relation
关系

=< is reflexive in the set of real number since for nay x we have x<= X similarly the relation of inclusion is reflexive in the family of all subsets of a universal set.
= <在实数集中是自反的，因为对于不存在x，我们有x <= X类似地，包含关系在通用集的所有子集的族中也是自反的。
A relation R is in a set X is symmetric if for every x and y in x whenever xRy then yRX that is R is a symmetric in x.
关系R是一组X是对称的，如果对于x中的每个x和y每当XRY然后YRX即R在X对称。

The relation
关系

<= and < are not symmetric i the set of real number while the relation of equality is.
<=和<不是对称i中的一组实数，而等式的关系。
A relation R in a set x is transitive if for every x, y and z in X whenever xRy and yRx then xRz that is R is transitive in X.
关系R中的一组X是传递的，若对所有的x，y和z在X每当XRY和YRX然后XRZ即R为传递在X。

The relation
关系

<= < and = are transitive in the set of real numbers. The relations and equality are also transitive in the family of a subset of a universal set.
<= <和=在实数集中是可传递的。关系和平等在通用集的子集的族中也是可传递的。