Indexed Sets Excercise

Let \{A_\alpha\}_{\alpha \in I} , and \{B_\alpha\}_{\alpha \in I} be two indexed families of subsets of a set S . Prove: \cup_{\alpha \in I}(A_\alpha \cup B_\alpha)= (\cup_{\alpha \in I}A_\alpha) \cup (\cup_{\alpha \in I} B_\alpha)

\forall_{x \in \cup_{\alpha \in I}(A_\alpha \cup B_\alpha)}(x \in A_\alpha or x \in B_\alpha) . This \Rightarrow that x \in \cup_{\alpha \in I}A_\alpha or x \in \cup_{\alpha \in I}B_\alpha \Rightarrow x \in (\cup_{\alpha \in I}A_\alpha) \cup (\cup_{\alpha \in I}B_{\alpha}) \Rightarrow \cup_{\alpha \in I}(A_\alpha \cup B_\alpha) \subset (\cup_{\alpha \in I}A\alpha) \cup (\cup_{\alpha \in I}B_\alpha).  

\forall_{x \in (\cup_{\alpha \in I}A_\alpha) \cup (\cup_{\alpha \in I}B_\alpha)}(x \in \cup_{\alpha \in I}A_\alpha or x \in \cup_{\alpha \in I}B_\alpha) \Rightarrow x \in \cup_{\alpha \in I}(A_{\alpha} \cup B_\alpha) \Rightarrow (\cup_{\alpha \in I}A_\alpha) \cup (\cup_{\alpha \in I}B_\alpha) \subset \cup_{\alpha \in I}(A_\alpha \cup B_\alpha) \therefore (\cup_{\alpha \in I} A_\alpha) \cup (\cup_{\alpha \in I}B_\alpha) = \cup_{\alpha \in I}(A_\alpha \cup B_\alpha ) .