## 2016年12月30日金曜日

### 数学 - 集合論 - 集合算 - 関係、同値関係、商集合(逆像)

$\begin{array}{l}a,b,c\in X\\ \\ f\left(a\right)=f\left(a\right)\\ ⇔f\left(a\right)Rf\left(a\right)\\ ⇔aR\text{'}a\\ \\ aR\text{'}b\\ ⇔f\left(a\right)Rf\left(b\right)\\ ⇒f\left(b\right)Rf\left(a\right)\\ ⇔bR\text{'}a\\ \\ aR\text{'}b\wedge bR\text{'}c\\ ⇔f\left(a\right)Rf\left(b\right)\wedge f\left(b\right)Rf\left(c\right)\\ ⇒f\left(a\right)Rf\left(c\right)\\ ⇔aR\text{'}c\end{array}$

$\begin{array}{l}f:X/R\text{'}\to Y/R\\ f\left({C}_{x}\right)=C{\text{'}}_{f\left(x\right)}\\ f:{C}_{x}\to C{\text{'}}_{f\left(x\right)}\\ \\ \\ 単射\\ {C}_{a},{C}_{b}\in X/R\text{'}\\ f\left(C{\text{'}}_{a}\right)=f\left(C{\text{'}}_{b}\right)\\ {C}_{f\left(a\right)}={C}_{f\left(b\right)}\\ f\left(a\right)R\text{'}f\left(b\right)\\ aR\text{'}b\\ C{\text{'}}_{a}=C{\text{'}}_{b}\\ \\ \\ fが全射の場合。\\ {C}_{y}\in Y/R\\ \exists x\in X\left[f\left(x\right)=y\right]\\ yRy\\ f\left(x\right)Rf\left(x\right)\\ xR\text{'}x\\ f\left({C}_{x}\right)={C}_{f\left(x\right)}={C}_{y}\\ \\ \\ X/R\text{'}\simeq Y/R\end{array}$