READING: I §2 5
Back to reading:
I §2 4
begin quote ita"
CHAPTER I [:] Notions from Set Theory [...] §2. OPERATIONS ON SETS. "end quote
These pages assume that the reader has the book and is reading along. topic index contents
Please read these notes from the beginning to here, and then paragraphs 8 and 9 of §2.
INDEXING FAMILIES OF SETS
If we are taking away the prerequisites for this material, then the equation below needs comment.
By convention, what is between the parentheses is done first. Consequently, from left to right, the three expressions mean (1) define the triple intersection as Rosenlicht has just done, (2) do the intersection first and then do the intersection with Z, and (3) do the intersection followed by the intersection with X.
It is interesting and useful to compare and contrast what we have here with what we would have if, instead of being sets, X, Y, and Z were numbers, and if, instead of doing intersections, we were doing addition or multiplication. We would need to pick two of the three numbers with one or the other going first into the calculator or onto paper. After doing this first addition or multiplication, we would then add or multiply with the third number. Similarly, with the intersection and union of sets, we get the same result no matter how we do the calculation. The contrast is that, with the indexing sets, we may not be able to order the elements of the indexing set - as we can order any number of numbers for which we may wish to find the sum or product. However, it seems to me that the details should be left until after chapter two. Then we will have the real number system, and we will be able to look at specific examples.
As an example of working with families of sets, problem 5 of chapter one will be solved.
Problem 5, chapter 1
Let I be a nonempty set and for each iI let be a set. Prove that
(a) for any set B we have
(a1): Recall that can be read as "implies" and that means "for all". Then:
(a2): With a slight shift of the wording, the second part is obtained from the first part by starting at the end and reversing the arrows:
(b) if each is a subset of a given set S, then
(b2): Here, we can simply start at the end of the argument in b1 and reverse the arrows.