Mathematics/Set Theory

Equivalence RelationDefinition 1. Let $R$ be a relation in a set $X$. Then we say that (a) $R$ is reflexive $\iff$ $\forall x \in X, xRx$.(b) $R$ is symmetric $\iff$ $xRy \Longrightarrow yRx$.(c) $R$ is transitive $\iff$ $xRy \wedge yRz \Longrightarrow xRz$. (d) $R$ is an equivalence relation $\iff$ $R$ is reflexive, symmetric, and transitive. Equivalence relation, 즉 동치 관계는 특정한 수학적 관점에서 볼 때 두 원소..

Partial OrderDefinition 1. A relation $\leq$ on a set $A$ is called a partial order relation if and only if the relation $\leq$ is reflexive and transitive on $A$ and antisymmetric on $A$, that is, if $a\leq b$ and $b \leq a$, then $a = b$. A partially ordered set is a pair $(A, \leq)$, where $A$ is a set and $\leq$ is a partial order relation on $A$.Total orderDefinition 2. A total order relati..

RelationDefinition 1. A relation $R$ from $A$ to $B$ is a subset of $A \times B$. It is customary to write $aRb$ for $(a, b) \in R$. The symbol $aRb$ is read $a$ is $R$-related to $b$.많은 경우 $A = B$이며, 이때 관계 $R$은 relation in $A$ 라고 말한다. Inverse RelationDefinition 2. Let $A, B$ be sets, not necessarily distinct, and let $R$ be a relation from $A$ to $B$. Then inverse $R^{-1}$ of $R$ is the relatio..