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Relationships between elements of sets are represented using the structure called
a relation.
9.1 Relations and Their Properties
A common relation is binary relation. Let $A$ and $B$ be sets. A binary relation
from $A$ to $B$ is a subset of $A X B$ 笛卡尔积. $a R b$ denote that $(a,b)$ belong
to $R$. In other word, $a$ is related to$b$ by $R$.
A function represcents a relation where exactly one element of $B$ is related t
each element of $A$. The graph of function $f$ from $A$ to $B$ is the set of ordered
pairs $(a,f(a))$ for $a$ belong to $A$. A relation on a set $A$ is a relation from
$A$ to $A$.
Qusetion: P 575 Example 4
Here are some properties. Relations $A$ relation $R$ on a set $A$ called reflexive
(反身的 if $(a,a)$ belong to $R$ for every elements $a$ belong $A$.
9.2 n-ary Relationsand Their Applications
9.3 Representing Rlations
9.4 Closure of Relations 闭包
9.5 Equivalence Relations
Definition 1: A relation on a set A is called an equivalence relation if
it is reflexive, symmetric, and transitive.
Definition 2: Two elements $a$ and $b$ that are related by an equivalence
relation are called equivalent. We mark it: $(a~b)$