離散數(shù)學(xué)關(guān)系的性質(zhì)

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(jìn)制運(yùn)算符，通常應(yīng)用于集合之間。 它是一組有序?qū)?#xff0c;其中該對(duì)的第一成員屬于第一集合，而該對(duì)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(guān)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(guān)系一詞表示一些熟悉的示例關(guān)系，例如父親與兒子的關(guān)系，母親與兒子的關(guān)系，兄弟與姐妹的關(guān)系等。算術(shù)中的常見示例是諸如“大于” ， “小于”或關(guān)系之間相等的關(guān)系。兩個(gè)實(shí)數(shù)。

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(duì)象對(duì)之間的稱為二進(jìn)制關(guān)系的關(guān)系。 在給出關(guān)系的集合理論定義之前，我們注意到可以通過將兩個(gè)對(duì)象按有序?qū)α谐鰜矶x兩個(gè)對(duì)象之間的關(guān)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?qū)Χ级x了二進(jìn)制關(guān)系。 我們將二元關(guān)系簡(jiǎn)稱為關(guān)系。 通過寫xRY可以表達(dá)特定的有序?qū)φf(x，y)ER的事實(shí)，其中R是一個(gè)關(guān)系，可以讀為“ x是R與y的關(guān)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(guān)系可以用集合(例如有序?qū)?描述，其中第一個(gè)成員是父親的名字，第二個(gè)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(shè)S為二元關(guān)系。 所有對(duì)象x的集合D(S)使得對(duì)于某個(gè)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(duì)象y的集合R(S) ，使得對(duì)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(guān)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(jìn)制關(guān)系的屬性

There are some properties of the binary relation:

二進(jìn)制關(guān)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(duì)于每個(gè)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(guān)系R是自反的。

The relation

關(guān)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(shí)數(shù)集中是自反的，因?yàn)閷?duì)于不存在x，我們有x <= X類似地，包含關(guān)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(guān)系R是一組X是對(duì)稱的，如果對(duì)于x中的每個(gè)x和y每當(dāng)XRY然后YRX即R在X對(duì)稱。

The relation

關(guān)系

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

<=和<不是對(duì)稱i中的一組實(shí)數(shù)，而等式的關(guān)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(guān)系R中的一組X是傳遞的，若對(duì)所有的x，y和z在X每當(dāng)XRY和YRX然后XRZ即R為傳遞在X。

The relation

關(guān)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(shí)數(shù)集中是可傳遞的。 關(guān)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(shù)學(xué)關(guān)系的性質(zhì)

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