question about indexing over sets

[proven dirge]

In a lot of the problems I've been doing I've been asked to let $I$ be a given arbitrary set and let ${A_i}_{i\in I}$ be a collection of sets. But this feels wrong because if we want to think of the indexing as a function $f:I\to S$ for some set $S$, don't we need to provide $S$ beforehand?

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[proven dirge]

What should that codomain be?

[limpid sparrow]

daniel_the_maniel

[proven dirge]

Obviously we need $S$ to be such that $f(I)\subseteq S$, but then we're referencing $f$ before it's been properly defined

[limpid sparrow]

daniel_the_maniel

[proven dirge]

Oh wait maybe it's ok since $f:I\to S$ is implicitly saying that $f(I)\subseteq S$

[limpid sparrow]

daniel_the_maniel

[quiet quest]

$S={A_i|i\in I}$ no?

[limpid sparrow]

Omegabet_

[quiet quest]

If you're given a collection of sets indexed by I, then a possible codomain is just.. the collection of sets

then $f:i\mapsto A_i$

[limpid sparrow]

Omegabet_

[proven dirge]

But bestie

You're referencing f before it's even been properly defined

Since the codomain is a part of the definition

Of a function

[quiet quest]

yeah, you have I and the collection of sets already

so just.. define the function

all that's left in defining a function, given a domain and codomain, is the rule between them

[proven dirge]

How do we get the collection of subsets

[quiet quest]

you were given it

[proven dirge]

Omg

[quiet quest]

{A_i} is a collection of sets

[proven dirge]

Wtf does A_i even mean tho

You have to imagine you have some function that takes in elements of I

And you use A_i as just another way of writing f(i)

[quiet quest]

define S as some sufficient power set then

You said you were given a collection of sets and wanted a function, so just send the index to the set indexed by said index

[solar copper]

The set of sets is defined along with the indexing on it

[proven dirge]

Doubt

[jagged kestrel]

rule of thumb

$\bigcup_{i\in I} A_i = \bigcup_{s_j \in I} \left( \bigcup_{i \in s_j} A_i \right)$

[limpid sparrow]

instantaneousCoffey

[jagged kestrel]

Notice that you can just distribute the index set into other families of sets too

now this goes on forever so you can keep asking for index sets.

${A_i} _{i\in I}$ read as : \ $A_i$ belong to the family when i belongs in the index set I

@proven dirge