離散數學關系的性質

笛卡爾積(A * B不等于B * A) (Cartesian product (A*B not equal to B*A))

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*表示的笛卡爾積是二進制運算符，通常應用于集合之間。 它是一組有序對，其中該對的第一成員屬于第一集合，而該對的第二成員屬于第二集合。

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)}

關系 (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.

關系一詞表示一些熟悉的示例關系，例如父親與兒子的關系，母親與兒子的關系，兄弟與姐妹的關系等。算術中的常見示例是諸如“大于” ， “小于”或關系之間相等的關系。兩個實數。

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.

在這里，我們將僅考慮對象對之間的稱為二進制關系的關系。 在給出關系的集合理論定義之前，我們注意到可以通過將兩個對象按有序對列出來定義兩個對象之間的關系。

Definition:

定義：

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".

任何一組有序對都定義了二進制關系。 我們將二元關系簡稱為關系。 通過寫xRY可以表達特定的有序對說(x，y)ER的事實，其中R是一個關系，可以讀為“ x是R與y的關系” 。

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:

父親與他的孩子的關系可以用集合(例如有序對)描述，其中第一個成員是父親的名字，第二個成員是他的孩子的名字，即：

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

F = {(x，y)| x是y的父親}

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.

設S為二元關系。 所有對象x的集合D(S)使得對于某個y ， (x，y)ES被稱為S的域。

范圍 (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.

所有對象y的集合R(S) ，使得對于某些x ， (x，y)ES表示為S的范圍。

Let r A B be a relation

令r AB為關系

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

集合中二進制關系的屬性

There are some properties of the binary relation:

二進制關系具有一些屬性：

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).

如果對于每個x EX ， xRx ，即(x，x)ER或R在X <==>(x)(x EX-> xRX)中是自反的，則集合X中的二元關系R是自反的。

The relation

關系

=< 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.

= <在實數集中是自反的，因為對于不存在x，我們有x <= X類似地，包含關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.

關系R是一組X是對稱的，如果對于x中的每個x和y每當XRY然后YRX即R在X對稱。

The relation

關系

<= and < are not symmetric i the set of real number while the relation of equality is.

<=和<不是對稱i中的一組實數，而等式的關系。

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.

關系R中的一組X是傳遞的，若對所有的x，y和z在X每當XRY和YRX然后XRZ即R為傳遞在X。

The relation

關系

<= < 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=在實數集中是可傳遞的。 關系和平等在通用集的子集的族中也是可傳遞的。

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

離散數學關系的性質

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