# Halmos-Generalized Version of Associative Law

1. Aug 30, 2012

### 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 {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.

Last edited: Aug 30, 2012
2. Aug 30, 2012

### mathman

Uk∈ KAk=Uj∈ J(Ui∈ IjAi)

Above expression is very unclear.

3. Aug 30, 2012

### sammycaps

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

4. Aug 31, 2012

### mathman

It looks like you are forming the union of Ak 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.