Most discussions of set theory with which I am familiar assign equal operator precedence to $\cap$ and $\cup$. This may sound like a ridiculous question, but I never thought about, nor do I recall encountering a formal rule regarding expressions of the form $\mathcal{F}=\mathcal{A}\cap\mathcal{B}\cup\mathcal{C}.$
Clearly $\mathcal{D}=\mathcal{A}\cap\left(\mathcal{B}\cup\mathcal{C}\right)$ is not equivalent to $\mathcal{E}=\left(\mathcal{A}\cap\mathcal{B}\right)\cup\mathcal{C}.$ The set $\mathcal{D}$ is necessarily a subset of $\mathcal{A};$ whereas $\mathcal{E}$ may contain an element of $\mathcal{C}$ which is not an element of $\mathcal{A}.$ So $\mathcal{E}$ is not necessarily a subset of $\mathcal{A}.$
I found myself confused while attempting Exercise 1-5.7(B) of Stoll's Set Theory and Logic, and finally realized that much of that confusion may be related to not having a formal rule for how to deal with the possibility of encountering an expression of the form of $\mathcal{F}.$
I haven't attempted to prove the proposition, but it seems evident that any statement using parentheses to preclude such ambiguity will always transform into another such statement when the rules of manipulation are correctly followed.
But what are we to do with an expression such as $\mathcal{F}?$ Reject it as logically ambiguous, and thus invalid? That seems to be the necessary choice. Is this correct?
You have to either reject it as ill-formed, or have in your system a rule assigning "$\cup$" and "$\cap$" unequal priorities. Really, though, this latter amounts to viewing "$A\cup B\cap C$" and similar expressions as abbreviations for expressions with parentheses placed appropriately. So there's no actual need to allow expressions like that at all, and I can't think why one would want to.
In particular, if you prove some logical result about all expressions which do involve parentheses as usual, it will also apply to expressions like "$A\cup B\cap C$" in the presence of an appropriate priority rule. I suspect that Stolle's exercise is working in a system where "$A\cup B\cap C$" is ill-formed, but there will be an analogue of the exercise which will work in any modification of that system to allow things like "$A\cup B\cap C$." So it won't make a substantive difference.