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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