READING: I §1 1
Back to reading:
begin quote ita"
CHAPTER I [:] Notions from Set Theory [...] §1. SETS AND ELEMENTS. SUBSETS. "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 1, 2, and 3 of §1.
EQUALITY OF SETS
Equality of sets is more subtle than it seems, and Rosenlicht says
begin quote ita-1-1-3-pg2 "
The statement that a set is completely determined by its elements may be written out as follows:
If X and Y are sets then X=Y if and only if, for all x, xX if and only if xY.
This time, the statement above made me stop and think long and hard. In my first reading of Rosenlicht's book, I passed over this quickly without giving it much thought. I think now that is why I had difficulty with some of the exercises - which just seemed too trivial. Now, I replace the words 'if and only if' by a double arrow, and I replace the words 'for all' by the upside down upper case letter 'A'. Then the condition which follows the part about X and Y being sets is
A statement which contains a double arrow should be read twice - once for each direction of the double arrow. When this is done, the arrows can be replaced by the word 'implies'. Try it now, but ignore the second double arrow. That is, treat what is on the right of the first double arrow as a kind of black box unit: X equals Y implies the condition on the right, and the condition on the right implies X equals Y.
Two Examples: (1) The picture below shows two seemingly different pieces of sheet music. Let set X contain just one element which is the ordered sequence of sounds produced by the score on the left. Let set Y contain just one element which is the ordered sequence of sounds produced according to the score on the right. Upon listening, we find that the sequence of sounds that is in X is also in Y and vice versa. So the condition is satisfied that is on the right of the first double arrow above. Then the left pointing part of that first double arrow tells us that the two sets are equal. (2) If we know that two sets are equal, then the right pointing part of the first double arrow above tells us that the condition on the right is satisfied so that 'being in one set' implies 'being in the other'.