Theorem. Suppose that $I$ and $J$ are two sets, and for all $\alpha \in I \cup J$ let $A_{\alpha}$ be a set. Show that $\bigcup\limits_{\alpha \in I}A_{\alpha} \cup \bigcup\limits_{\alpha \in J} A_{\alpha} = \bigcup\limits_{\alpha \in I \cup J} A_\alpha$. If $I$ and $J$ are non-empty, show that $\bigcap\limits_{\alpha \in I}A_{\alpha} \cap \bigcap\limits_{\alpha \in J} A_{\alpha} = \bigcap\limits_{\alpha \in I \cup J} A_\alpha$.
Proof.

(i) $$y \in \left( \bigcup\limits_{\alpha \in I}A_{\alpha} \cup \bigcup\limits_{\alpha \in J} A_{\alpha} \right) \leftrightarrow \left( y \in \bigcup\limits_{\alpha \in I}A_{\alpha} \right) \vee \left( y \in \bigcup\limits_{\alpha \in J} A_{\alpha} \right) \leftrightarrow \left( (\exists \alpha \in I) \: y \in A_{\alpha} \right) \vee \left( (\exists \alpha \in J) \: y \in A_{\alpha} \right) \leftrightarrow ( y \in A_{i_1} \vee y \in A_{i_2} \vee ... ) \vee ( y \in A_{j_1} \vee y \in A_{j_2} \vee ... )$$ $$\leftrightarrow y \in A_{i_1} \vee y \in A_{j_1} \vee ... \leftrightarrow (\exists \alpha \in I \cup J) \: y \in A_{\alpha} \leftrightarrow y \in \bigcup\limits_{\alpha \in I\cup J} A_{\alpha}.$$ Two sets are equal iff every element of one set is an element of the other and vice versa: $$\left( y \in \bigcup\limits_{\alpha \in I}A_{\alpha} \cup \bigcup\limits_{\alpha \in J} A_{\alpha} \leftrightarrow y \in \bigcup\limits_{\alpha \in I \cup J} A_\alpha \right) \leftrightarrow \left( \bigcup\limits_{\alpha \in I}A_{\alpha} \cup \bigcup\limits_{\alpha \in J} A_{\alpha} = \bigcup\limits_{\alpha \in I \cup J} A_\alpha \right).$$
(ii) $$y \in \left( \bigcap\limits_{\alpha \in I}A_{\alpha} \cap \bigcap\limits_{\alpha \in J} A_{\alpha} \right) \leftrightarrow \left( y \in \bigcap\limits_{\alpha \in I}A_{\alpha} \right) \wedge \left( y \in \bigcap\limits_{\alpha \in J} A_{\alpha} \right) \leftrightarrow \left( (\forall \alpha \in I) \: y \in A_{\alpha} \right) \wedge \left( (\forall \alpha \in J) \: y \in A_{\alpha} \right) \leftrightarrow ( y \in A_{i_1} \wedge y \in A_{i_2} \wedge ... ) \wedge ( y \in A_{j_1} \wedge y \in A_{j_2} \wedge ... )$$ $$\leftrightarrow y \in A_{i_1} \wedge y \in A_{j_1} \wedge ... \leftrightarrow (\forall \alpha \in I \cup J) \: y \in A_{\alpha} \leftrightarrow y \in \bigcap\limits_{\alpha \in I\cup J} A_{\alpha}.$$ Two sets are equal iff every element of one set is an element of the other and vice versa: $$\left( y \in \bigcap\limits_{\alpha \in I}A_{\alpha} \cap \bigcap\limits_{\alpha \in J} A_{\alpha} \leftrightarrow y \in \bigcap\limits_{\alpha \in I \cup J} A_\alpha \right) \leftrightarrow \left( \bigcap\limits_{\alpha \in I}A_{\alpha} \cap \bigcap\limits_{\alpha \in J} A_{\alpha} = \bigcap\limits_{\alpha \in I \cup J} A_\alpha \right).$$ $\Box$
2015 FEB 01 (v1.0)
Contact me: m.herrmann followed by an -at- followed by blaetterundsterne.org.

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