Kartezyen Çarpım-V

Question

$A$ ve her  $\alpha\in \Lambda$  için  $B_{\alpha}$  herhangi kümeler olmak üzere

$$A\times \left(\bigcap_{\alpha\in \Lambda} B_{\alpha} \right)=\bigcap_{\alpha\in \Lambda}\left(A\times B_{\alpha} \right)$$ olduğunu gösteriniz.

Answer 1

$$(x,y)\in A\times\left(\bigcap_{\alpha\in\Lambda}B_{\alpha}\right)$$ $$\Leftrightarrow$$ $$x\in A\wedge y\in \left(\bigcap_{\alpha\in\Lambda}B_{\alpha}\right)$$ $$\Leftrightarrow$$ $$x\in A \wedge (\forall\alpha\in\Lambda)(y\in B_{\alpha})$$ $$\overset{?}{\Leftrightarrow}$$ $$(\forall\alpha\in\Lambda)(x\in A \wedge y\in B_{\alpha})$$ $$\Leftrightarrow$$ $$(\forall\alpha\in\Lambda)((x,y)\in A\times B_{\alpha})$$ $$\Leftrightarrow$$ $$(x,y)\in\bigcap_{\alpha\in\Lambda}(A\times B_{\alpha}).$$