Halmos-Generalized Version of Associative Law

[sammycaps]

I posted this in the homework section, but I think it probably belongs here.

So Halmos says in Section 9 on families, "Suppose, for instance, that {I_j} is a family of sets with domain J, say; write K=U_jI_j and let {A_k} be a family of sets with domain K. Is it then not difficult to prove that, U_{k∈ K}A_k=U_{j∈ J}(U_{i∈ Ij}Ai); this is the generalized version of the associative law for unions

So, I'm just trying to wrap my head around why this is the generalized version of the associative law for unions. Do we have to assign some kind of sequence for the way we take the union over K that somehow transfers to the way we take union over J and Ij? This may be a dumb question, but I'm a bit confused.

 

[mathman]

I posted this in the homework section, but I think it probably belongs here.

So Halmos says in Section 9 on families, "Suppose, for instance, that {Ij} is a family of sets with domain J, say; write K=UjIj and let {Ak} be a family of sets with domain K. Is it then not difficult to prove that, Uk∈ KAk=Uj∈ J(Ui∈ IjAi); this is the generalized version of the associative law for unions

So, I'm just trying to wrap my head around why this is the generalized version of the associative law for unions. Do we have to assign some kind of sequence for the way we take the union over K that somehow transfers to the way we take union over J and Ij? This may be a dumb question, but I'm a bit confused.

Uk∈ KAk=Uj∈ J(Ui∈ IjAi)

Above expression is very unclear.

 

[sammycaps]

Oh, my bad, when I copy pasted, the formatting was lost in translation. Fixed.

 

[mathman]

It looks like you are forming the union of A_k on both sides of the equation, but in a different order on each side. On the left (since there are no parentheses), the order seems to be one at a time, while on the right you are taking unions of a bunch at a time and then union of the bunches.

 

[blue_raver22]

I would respond by saying that the generalized version of the associative law for unions is a mathematical concept that allows us to simplify and manipulate sets and their unions in a more abstract and general way. It may seem confusing or counterintuitive at first, but once we understand the concept and its applications, it can be a powerful tool in solving problems and proving theorems.

In this case, Halmos is showing that the order in which we take unions over sets does not matter, as long as the sets are related through a common index set. This means that we can rearrange the unions in any way we want, as long as the elements in the sets are the same. This is similar to the associative law for addition and multiplication, where the order of operations does not affect the final result.

To answer your question, we do not necessarily need to assign a specific sequence for taking unions over K, but rather understand the relationship between the sets and how they can be grouped and manipulated. This generalization allows us to apply the associative law for unions in a broader context, beyond just two sets.

In summary, the generalized version of the associative law for unions is a powerful tool in mathematics that allows us to manipulate and simplify sets and their unions in a more abstract and general way, without being limited to just two sets. I hope this helps clarify the concept for you.