READING: I §1 8
Back to reading:
I §1 7
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 1 and 2 of §2.
UNIONS AND INTERSECTIONS OF SETS
We have the union and intersection of two sets:
Note that the condition for the union uses the word 'or' and the symbol for union, , looks like an upper case U. The intersection symbol can then be remembered as the union symbol turned upside down. Similaly below, the word 'or' will be represented by the symbol , and the symbol for 'and' can be remembered by turning the or symbol upside down.
SOME MATHEMATICAL & SYMBOLIC LOGIC
Beyond the use of symbols, I use the term symbolic logic becuase my source is the book Symbolic Logic by Copi. I will work from memory because the book is in storage. By the term mathematical logic, I mean a certain restriction of the meaning of statements of the form 'If A, then B'. Copi calls this restriction material implication.
Begin by letting capital letters like A and B represent statements that must be either true or false. Let the negation of these statements be represented by ~A and ~B, which we read as 'not A' and 'not B'. Then we can create truth tables like the one shown below. It shows both possible truth possibilities for the statement A, and the resulting truth values for ~A.
Following Rosenlicht's presentation of the meaning of the word 'or' in mathematics, we see that the statement 'A or B' has the following truth table:
The statement 'A and B' has the following truth table:
Now we come to Copi's discussion of the statement 'If A, then B.' In ordinary language this can have several senses. One is that of definition, which Rosenlicht uses and which seems actually to be impossible to avoid: If a polygon has three sides, then it is a triangle. Another sense is one of physical causation: If a pinch of sugar is added to a hot mug of coffee, then the sugar will dissolve. Another sense is rhetorical: If so-and-so means well, then I am a hobbit. This last one is almost literary and is very non-mathematical. Copi points out that the one thing all the senses have in common is that the statement 'if A then B' is false if A is true and B is false. Therefore, mathematicians agree that, in mathematical logic, the entire meaning of 'if A then B' will be , and we represent both 'if A then B' and the symbol just given by . The truth table is ...
... and each column after the second is filled in using the information in the previous columns. The important fact for understanding that the empty set is a subset of every set is that is always true when A is false!
Recall that we had ...
begin quote ita-1-1-6-pg3 "
[...] is shorthand for the statement "if x X, then x Y".
If X is the empty set, and Y is any other set, then means if x then xY, and this is always true because the 'if' part is false for all x.