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.