If we are talking about interface like in Java, for instance, then the important difference is that in Java, class specifies what interfaces it has, while in Haskell, any code can specify how a type is a member of a type class (actually there can be multiple different ways how the same type can be member of same type class). Another difference was, until recently, that the type classes can define "default" implementations of functions. And there are other differences related to "deriving" statement.
> The terminology "class" in Haskell certainly has nothing to do with its mathematical meaning of "collection of sets defined by a predicate"
I don't want to get into philosophy of math, I am not strong in it, but "class" in mathematics is certainly not limited to collections of sets. For example, there is a class of all classes, but not set of all sets.
The predicate that defines the type class in Haskell is defined by the collection of all the relevant "instance" statements.
Actually, now that I think about it, as I note above, there are multiple ways that one type can be made a member of one type class, so it's certainly not a membership in the set-theoretic sense.
If we are talking about interface like in Java, for instance, then the important difference is that in Java, class specifies what interfaces it has, while in Haskell, any code can specify how a type is a member of a type class (actually there can be multiple different ways how the same type can be member of same type class). Another difference was, until recently, that the type classes can define "default" implementations of functions. And there are other differences related to "deriving" statement.
> The terminology "class" in Haskell certainly has nothing to do with its mathematical meaning of "collection of sets defined by a predicate"
I don't want to get into philosophy of math, I am not strong in it, but "class" in mathematics is certainly not limited to collections of sets. For example, there is a class of all classes, but not set of all sets.
The predicate that defines the type class in Haskell is defined by the collection of all the relevant "instance" statements.
Actually, now that I think about it, as I note above, there are multiple ways that one type can be made a member of one type class, so it's certainly not a membership in the set-theoretic sense.