Indexed unions and definitions

Question

I've been trying to understand how to explicitly state an indexed union/intersection of sets. Here is an example:

$X$ $=$ $\bigcup S_\alpha$ where $\alpha \in I$

First of all, how would one read this? Would it be the union of all sets $S$ that like $S_a$, $S_b$, $S_c$ etc where $a,b,c \in I$ ?

So, if I were to explicitly state $X$ would be $X$ $=$ {$x \in S_\alpha :$ (not sure what goes here)}

And what about the intersection?

Help would be greatly appreciated!

Answer 1

Suppose $I$ is a set and you have a collection $\{ S_{\alpha}\, :\, \alpha \in I \}$ of sets indexed by $I$. For example, if $I=\{1,2,3\}$ then your collection of sets would be $\{ S_1, S_2, S_3 \}$. Then we define the union of all the $S_{\alpha}$ for $\alpha \in I$ by: $$\bigcup_{\alpha \in I} S_{\alpha} = \{ x\, :\, x \in S_{\alpha}\ \text{for some}\ \alpha \in I \}$$ It's the set of all those things that appear in at least one of the $S_{\alpha}$s. So for example if $I=\{1,2,3\}$ then $$\bigcup_{\alpha \in I} S_{\alpha} = \bigcup_{\alpha=1}^3 S_{\alpha} = S_1 \cup S_2 \cup S_3$$

The intersection is similar: $$\bigcap_{\alpha \in I} S_{\alpha}= \{ x\, :\, x \in S_{\alpha}\ \text{for all}\ \alpha \in I \}$$

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As for how you read it, that's up to you really, as long as it's clear. Common ways include:

• the union over all $\alpha$ in $I$ of $S_{\alpha}$
• the union of all the $S_{\alpha}$ for $\alpha$ in $I$
• the union of all the $S_{\alpha}$s $\qquad \leftarrow$ if $I$ is clear from context

Answer 2

Yes! The notation $$X=\bigcup_{\alpha\in I} S_\alpha$$ defines $$X=\{x:\exists \alpha\in I:x\in S_\alpha\}$$

Similarly, $$X=\bigcap_{\alpha\in I} S_\alpha = \{x:\forall\alpha\in I:x\in S_\alpha\}$$