How do you read set union and intersection notation? How do you read
$$A \bigcup B $$
"The union of A and B" (starting with the operator) is unintuitive ("x plus y" is more intuitive and shorter than "the addition of x and y").  Furthermroe, in a long statement the operator may be more inward when reading from left to right.  "A or B" feels unclear.  I have heard people say "A union B", which feels more natural, but is grammatically questionable.
Similarly, how do you read
$$\bigcup_{i\epsilon I}^{} A_{i}$$
"The union of all (sets) $A_i$, where $i$ is in $I$"?  "The union, where $i$ is in $I$, of all (sets) $A_i$"?  The first sounds more natural, but goes from the operator to the argument and then back to the operator.  Is there a correct way?
Thanks in advance.
 A: Generally, there are not fixed rules on how to turn mathematical notation into spoken words.  Indeed, one of the most important features of mathematical notation is its ability to represent complicated concepts compactly, and as a result it often cannot be easily translated into a linear flow of words without complicated circumlocutions.  So, you should feel free to pronounce mathematical notations like these in any way you feel is natural and conveys the meaning clearly.
That said, by far the most common and standard pronunciation of $A\cup B$ is "$A$ union $B$".  As you say, when considered as a string of ordinary words, it may seem dubiously grammatical, but it's a completely acceptable thing to say in spoken mathematics.
For $\bigcup_{i\in I}A_i$, there is not any specific common pronunciation.  The two that you propose are both fine and most people would usually say something similar to them, but there are many different variations.  I would personally probably say something like "the union of the $A_i$ over all $i$ in $I$".
