According to a book I'm reading (A Set Theory Workbook by Iain T. Adamson), if a is a set and belongs to a class B, then the union of B is a subset of a (at least, according to the book, this is provable in NBG).
My question is, how could this be?
Let's say that a = {1, 2}, b = {2, 3}, and B = {a, b}. My understanding is that the union of B = {1, 3}; thus, the union of B cannot be a subset of a because the number "3" does not belong to a.
Yet here's a quote from the book I'm reading: "Suppose a belongs to B. Let x be any element of a. Since x belongs to a and a belongs to B it follows that x belongs to the union of B. So a is a subset of the union of B (page 79).
What am I missing here?
The union is the set that includes all that is common amongst the unioned sets. One way that helped me understand it was to think of it as dissolving the sets. In your example the union of B would be the union of the set
Notice how we keep the outer brackets but dissolve the brackets for 'a' and 'b'. Thus, we keep all that is common amongst 'a' and 'b'. However, {1, 2, 3} is not a subset of {1, 2}. It is the other way around.
I should probably rephrase what I said initially, because an intersection can be stated in a similar (and thus confusing) form: i.e., the intersection is what is common to each set. Therefore, the intersection of S, for some S = {a, b, c, ...} is what is common to each set a, b, c, ... In my initial phrasing, I am saying what is in the union of S is what is common among a, b, c; to be in the union is to be in any of a, b, c, ... Therefore, to rephrase, I should say the union is the set that includes all that is among the unioned sets. I believe this way better reflects the intuitive approach I had about dissolving the sets in the union, so that you are left with all that is among those sets.
I could raise the philosophical point that one can only know definitions, not understand them, for there is nothing in them to understand. Definitions portray ideas, and it is in the ideas of our terms that we come to an understanding. For instance, one can know the algebraic definition of a dot product without having any understanding of the idea it conveys geometrically, which anyone can argue is the root of its very inception. However, to make such a point is to travel into needless pedantry. The point of elucidating the idea of a union on sets was merely to convey an intuitive comprehension I obtained, and that others I know have found fruitful. The more lay approach to conveying such an idea is so that one can have familiarity with the idea beyond just symbolic representation and perceiving its instances. If that leads one to better grasp its use, is that not learning and understanding the definition?