Hi all. In chapter 9 of Halmos's book titled Naive set theory, he talks about families of sets. He then talks about the associativity of sets as follows(adsbygoogle = window.adsbygoogle || []).push({});

"The algebraic laws satisfied by the operation of union for pairs can be

generalized to arbitrary unions. Suppose, for instance, that {I_{j}} is a

family of sets with domain J, say; write K = [itex]\bigcup[/itex]_{j}I_{j}and A_{k}be a family

of sets with domain K. It is then not difficult to prove that

[itex]\bigcup[/itex]_{k[itex]\in[/itex]K}A_{k}= [itex]\bigcup[/itex]_{j[itex]\in[/itex]J}([itex]\bigcup[/itex]_{i[itex]\in[/itex]Ij}A_{i});

this is the generalized version of the associative law for unions. Exercise:

formulate and prove a generalized version of the commutative law".

I could prove the associative law; however, I cannot formulate the commutative law. Basically formulating the generalized commutativity of union of sets in terms of indexed family of sets is not clear to me. How will the commutative law look like when written using the jargon of indexed family of sets? I would greatly appreciate your help in this matter.

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# Associativity and commutativity of sets

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