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I'm learning about indexed sets and came across the definition of an indexed set to be, "...any set,such that for $i \in I$, we have a set $A_i$ ...how would this definition be written using quantifiers? Please let me know if this definition is wrong as well...thank you!

I think it should look like this? but I'm not sure:

$$ \underset{i \in I}{\forall} \exists A_i $$

HallaSurvivor
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NotBadG
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  • Welcome to MSE! I've edited your question to use mathjax, which is searchable, rather than an image, which is not. This way future users of the site will have an easier time finding this question. In the future, you should do the same ^_^ – HallaSurvivor Feb 28 '22 at 08:13
  • $\forall i \in I \exists A_i$ is not a definition of anything. It is a trivially true sentence of first-order logic which merely states that for each $i \in I$ some set exists. This is true simply because, yes, some set exists (for example, the empty set). – Pilcrow Feb 28 '22 at 11:35

1 Answers1

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Your idea is right, even if your notation (setting the $i \in I$ underneath the $\forall$) looks a bit strange to my eyes.

If we want to codify the idea that "for each $i \in I$ there is a set $A_i$", then we would write

$$ \forall i \in I . \exists A_i $$

As an aside, this can be "implemented" in any number of ways.

Perhaps most naturally as function $A_{(-)} : I \to \mathsf{Set}$ which takes each element $i$ to the set $A_i$, but (for technical reasons) we often try to avoid functions into $\mathsf{Set}$ (since the set of all sets doesn't exist).

This leads to a related, if more difficult to motivate, implementation (called a "bundle") where we view the family as a single set $A$ with a map $p : A \to I$. Then we write $A_i = p^{-1}(i)$. This feels backwards, but turns out to be a very useful perspective to have on indexed sets!

I hope this helps ^_^

HallaSurvivor
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  • thank you! your response was very helpful... I'd like to read up a bit more about index sets...if you know any good resources (books or websites) please let me know...thanks again – NotBadG Feb 28 '22 at 10:54