How do I interpret this set builder notation into English terms?

Question

How would I interpret this set builder notation into English terms?

If $A_\alpha$ is a set for every $\alpha$ in some index set $I\ne\emptyset$,$$\begin{align}\bigcup_{\alpha\in I}A_\alpha&=\{x\::\:x\in A_\alpha\text{ for at least one set}A_\alpha\text{ with }\alpha\in I\}\\\bigcap_{\alpha\in I}A_\alpha&=\{x\::\:x\in A_\alpha\text{ for every set }A_\alpha\text{ with }\alpha\in I\}.\end{align}$$

I thought of saying something similar to,"The Set of All $x$ in $A$ for at least one set $A$ with the set of all $a$ in $I$.", but that doesn't sound correct

Answer 1

Your examples are already partly in English, as opposed to definitions such as$$\bigcup_{\alpha\in I}A_\alpha=\{x\::\:\exists\alpha\in I(x\in A_\alpha)\},\,\bigcap_{\alpha\in I}A_\alpha=\{x\::\:\forall\alpha\in I(x\in A_\alpha)\}.$$But it's a spectrum. So if we try to go even further, I'd translate them respectively as:

• The union of some sets (in this case the $A_\alpha$) is the set of things that are elements of at least one of them;
• The intersection of some sets is the set of things that are elements of all of them.