I have an exercise book from my university which doesn't specify a quantifier.
It uses expressions like "here $A$,$B$,$C$ are sets", or "if $x \notin A$ then ..." (it uses $x$ before it is even defined, just out of nowhere).
I'm going to need an answer from my university, of course, but I want to ask in general context: Is there any implicit quantifier when one is not specified, or is it always an error? If there is an implicit quantifier, which is it?
On
Generally speaking it is very, very common in mathematics to write a natural language statement like "if $x \in X$ then BLAH BLAH" which then translates into the more explicitly quantified statement "for all $x \in X$ BLAH BLAH". There are many, many other dictionary entries which translate between natural language mathematical statements and explicitly quantified statements. You can probably encounter many such dictionary entries in any entry level calculus textbook, where formal quantification is usually treated on an intuitive level.
I, personally, consider it very bad practice to not quantify every variable. Mathematicians are divided in this respect and one can even note that mathematicians from certain fields do this kind of thing more often than mathematicians in other fields.
Basically it's a matter of opinion whether it is incorrect or not.
Usually, if a quantifier is missing, the variable is assume to be quantified universally.