Let $$ I $$ and $$ J $$ be arbitrary index sets (either empty, finite, countable or uncountable) and let $$ \left(A_{i}\right)_{i\in I} $$ and $$ \left(B_{j}\right)_{j\in J} $$ be tuples of arbitrary sets. Is it true that:
$$ \bigcup_{(i,j)\in I\times J}(A_{i}\times A_{j})=\left(\bigcup_{i\in I}A_{i}\right)\times\left(\bigcup_{j\in J}B_{j}\right) $$?
Please be brief.

Yes, the statement is **true**. Here's why:

For any $(i, j) \in I \times J$, the set $ A_i \times B_j $ is a subset of $ \left( \bigcup_{i \in I} A_i \right) \times \left( \bigcup_{j \in J} B_j \right) $. Thus, the union of all such pairs $ A_i \times B_j $ over $ I \times J $ will be exactly equal to the Cartesian product of the unions.

Formally:
\[
\bigcup_{(i,j) \in I \times J} (A_i \times B_j) = \left( \bigcup_{i \in I} A_i \right) \times \left( \bigcup_{j \in J} B_j \right)
\]

Would you like further clarification or examples?

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Here are 5 follow-up questions related to set theory and Cartesian products:
1. What is the Cartesian product of two sets and how is it constructed?
2. How do Cartesian products behave with respect to intersections?
3. What changes if one or both index sets, $ I $ or $ J $, are empty?
4. Can you apply this result to finite or countable sets specifically?
5. How does this result extend to arbitrary products of sets (indexed by a general family)?

**Tip:** When working with unions and Cartesian products, visualize them as combining elements from two sets in pairs.

Let $$ I $$ and $$ J $$ be arbitrary index sets (either empty, finite, countable or uncountable) and let $$ \left(A_{i}\right)_{i\in I} $$ and $$ \left(B_{j}\right)_{j\in J} $$ be tuples of arbitrary sets. Is it true that: $$ \bigcup_{(i,j)\in I\times J}(A_{i}\times B_{j})=\left(\bigcup_{i\in I}A_{i}\right)\times\left(\bigcup_{j\in J}B_{j}\right) $$? Please be brief.

Explore the proof and explanation of the distributive law for Cartesian products and unions of indexed sets. Learn how set theory concepts like Cartesian products, unions, and indexed families of sets interact in this detailed mathematical explanation.

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