Relations and Their Basic Properties
Relations and Their Basic Properties
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关系及其基本属性
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发表时间：
2004
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通讯作者：
Edmund Woronowicz
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作者：
Edmund Woronowicz
我们在这里定义：模式关系作为一组对，域，上域和关系域;空和单位关系，关系的合成，关系下的集合的图像和逆图像。重新定义了两个谓词=和，以及三个函数、和\。介绍了上述概念的基本事实。文章[2]和[1]提供了本文的符号和术语。让我1是一个集合。我们说I1是类关系的当且仅当：（定义1）如果x ∈ I1，则存在y，z使得x = y，z。让我们注意到存在一个类关系的空集合。二元关系是一个类关系集合。表示二元关系。我们现在陈述四个命题：（3）1如果A ≠ R，则A是类关系的。(4){{ x，y }是关系型的。Rel存在方案处理集合A、B和二元谓词P，并声明：存在二元关系R使得对于所有x，y保持x，y ∈ R当且仅当x ∈ A且y ∈ B且P [x，y]对于所有参数值。让我们考虑P，R。设P = R当且仅当：（定义2）对所有的a，B保持a，B ∈ P当且仅当a，B ∈ R。[1]命题（1）和（2）已被删除。
We define here: mode Relation as a set of pairs, the domain, the codomain, and the field of relation; the empty and the identity relations, the composition of relations, the image and the inverse image of a set under a relation. Two predicates, = and ⊆, and three functions, ∪, ∩ and \ are redefined. Basic facts about the above mentioned notions are presented. The articles [2] and [1] provide the notation and terminology for this paper. Let I 1 be a set. We say that I 1 is relation-like if and only if: (Def. 1) If x ∈ I 1 , then there exist y, z such that x = y, z. Let us note that there exists a set which is relation-like and empty. A binary relation is a relation-like set. denote binary relations. We now state four propositions: (3) 1 If A ⊆ R, then A is relation-like. (4) {{ x, y } is relation-like. The scheme Rel Existence deals with sets A, B and a binary predicate P , and states that: There exists a binary relation R such that for all x, y holds x, y ∈ R iff x ∈ A and y ∈ B and P [x, y] for all values of the parameters. Let us consider P, R. Let us observe that P = R if and only if: (Def. 2) For all a, b holds a, b ∈ P iff a, b ∈ R. 1 The propositions (1) and (2) have been removed.