## 2016年11月11日金曜日

### 数学 - 集合と写像 - 添数づけられた族、一般の直積(色々な等式)

1. $\begin{array}{l}x\in \left(\underset{\lambda \in \Delta }{\cup }{A}_{\lambda }\right)\cap \left(\underset{\mu \in M}{\cup }{B}_{\lambda }\right)\\ ⇔x\in \underset{\lambda \in \Delta }{\cup }{A}_{\lambda }\wedge x\in \underset{\mu \in M}{\cup }{B}_{\lambda }\\ ⇔\exists \lambda \in \Delta \left[x\in {A}_{\lambda }\right]\wedge \exists \mu \in M\left[x\in {B}_{\mu }\right]\\ ⇔\exists \lambda \in \Delta \exists \mu \in M\left[x\in {A}_{\lambda }\cap {B}_{\mu }\right]\\ ⇔x\in \underset{\left(\lambda ,\mu \right)\in \Delta ×M}{\cup }\left({A}_{\lambda }\cap {B}_{\mu }\right)\end{array}$

2. $\begin{array}{l}x\in \left(\underset{\lambda \in \Delta }{\cap }{A}_{\lambda }\right)\cup \left(\underset{\mu \in M}{\cap }{B}_{\mu }\right)\\ ⇔x\in \underset{\lambda \in \Delta }{\cap }{A}_{\lambda }\vee x\in \underset{\mu \in M}{\cap }{B}_{\mu }\\ ⇔\forall \lambda \in \Delta \left[x\in {A}_{\lambda }\right]\vee \forall \mu \in M\left[x\in {B}_{\mu }\right]\\ ⇔\forall \lambda \in \Delta \forall \mu \in M\left[x\in {A}_{\lambda }\vee x\in {B}_{\mu }\right]\\ ⇔\forall \lambda \in \Delta \forall \mu \in M\left[x\in {A}_{\lambda }\cup {B}_{\mu }\right]\\ ⇔x\in \underset{\left(\lambda ,\mu \right)\in \Delta ×M}{\cap }\left({A}_{\lambda }\cup {B}_{\mu }\right)\end{array}$

3. $\begin{array}{l}\left(x,y\right)\in \left(\underset{\lambda \in \Delta }{\cup }{A}_{\lambda }\right)×\left(\underset{\mu \in M}{\cup }{B}_{\mu }\right)\\ x\in \underset{\lambda \in \Delta }{\cup }{A}_{\lambda }\wedge y\in \underset{\mu \in M}{\cup }{B}_{\mu }\\ \exists \lambda \in \Delta \left[x\in {A}_{\lambda }\right]\wedge \exists \mu \in M\left[y\in {B}_{\mu }\right]\\ \exists \lambda \in \Delta \exists \mu \in M\left[\left(x,y\right)\in {A}_{\lambda }×{B}_{\mu }\right]\\ \left(x,y\right)\in \underset{\left(\lambda ,\mu \right)}{\cup }\left({A}_{\lambda }×{B}_{\mu }\right)\end{array}$

4. $\begin{array}{l}\left(x,y\right)\in \left(\underset{\lambda \in \Delta }{\cap }{A}_{\lambda }\right)×\left(\underset{\mu \in M}{\cap }{B}_{\mu }\right)\\ x\in \underset{\lambda \in \Delta }{\cap }{A}_{\lambda }\wedge y\in \underset{\mu \in M}{\cap }{B}_{\mu }\\ \forall \lambda \in \Delta \left[x\in {A}_{\lambda }\right]\wedge \forall \mu \in M\left[y\in {B}_{\mu }\right]\\ \forall \lambda \in \Delta \forall \mu \in M\left[x\in {A}_{\lambda }\wedge y\in {B}_{\mu }\right]\\ \forall \lambda \in \Delta \forall \mu \in M\left[x\in {A}_{\lambda }\cap {B}_{\mu }\right]\\ \left(x,y\right)\in \underset{\left(\lambda ,\mu \right)\in \Delta ×M}{\cap }\left({A}_{\lambda }\cap {B}_{\mu }\right)\end{array}$