Chapter 11: equivalence relation

等價關係 equivalence relation

\[\begin{align*} & R\text{ is an equivalence relation over }A\times B\\ \Leftrightarrow & \begin{cases} R=\sim=\left\{ \left\langle x,y\right\rangle \middle|x\sim y\right\} \subseteq A\times B & \left(\text{e}\right)\text{equivalence 等價}\\ \vdots & \vdots \end{cases}\\ \Leftrightarrow & \begin{cases} R=\left\{ \left\langle x,y\right\rangle \middle|xRy\right\} \subseteq A\times B & \left(R\right)\text{relation}\\ \forall\left\langle x,y\right\rangle \in R\left(xRx\right) & \left(r\right)\text{reflexive}\\ \forall\left\langle x,y\right\rangle \in R\left(xRy\Rightarrow yRx\right) & \left(s\right)\text{symmetric}\\ \forall\left\langle x,y\right\rangle ,\left\langle y,z\right\rangle \in R\left(\begin{cases} xRy\\ yRz \end{cases}\Rightarrow xRz\right) & \left(t\right)\text{transitive} \end{cases}\Leftrightarrow\begin{cases} R=\left\{ \left\langle x,y\right\rangle \middle|xRy\right\} \subseteq A\times B & \text{關係}\\ \forall\left\langle x,y\right\rangle \in R\left(\left\langle x,x\right\rangle \in R\right) & \text{自反}\\ \forall\left\langle x,y\right\rangle \in R\left(\left\langle y,x\right\rangle \in R\right) & \text{對稱}\\ \forall\left\langle x,y\right\rangle ,\left\langle y,z\right\rangle \in R\left(\left\langle x,z\right\rangle \in R\right) & \text{遞移} \end{cases} \end{align*}\]