Not that I expect anyone to read it...
But- perhaps someday I will manage to write something interesting enough that even my most trivial thoughts will be pored over by generations of scholars!
I am ever the optimist.
Still, this is a digression, I titled this blog 'Explorations' to indicate that I, as the author, would be exploring various things -- most notably mathematics (hence, mathexplorations...). So- to begin at the beginning of all math that I have seen:
Set Theory
Halmos' book, "Naive Set Theory" points out that every professional mathematician should have at least a little knowledge of set theory, and every upper division and beginning graduate course seems to start with a quick review of set-theory concepts.
Naive set theory starts us off, and is enough to cover most everything you need at lower levels. To me, it is merely an extension of logical statements:
"All A are B." is equivalent to saying, "The set of all A is a subset of all B" or "A implies B". At least these statements are all related in a strong sense... Notice that "A implies B" is often restated as "If A, then B", but that has a causal connotation that is not present in the logical statement.
"a is an A." is likewise equivalent to, "a is an element of the set A"
To use the canonical example, "All men are mortal. Socrates is a man. Therefore Socrates is mortal." is the same as saying, "The set of all men is a subset of all things mortal. 'Socrates' is an element of the set of all men, therefore 'Socrates' is an element of all things mortal."
"Or- if Socrates is a man, and if being a man implies one is mortal, then Socrates is mortal.
Of course- it sounds better in the original...
It is important that these definitions not give rise to any contradictions. We might not be able to work everything out, but that which we can work out, we know to be correct!
So- we start off by declaring that we have sets and elements of sets. We define equality of sets to be when they contain the same elements and look at the standard set operations of:
Union or (in A or B or maybe both)
Intersection and (in A and B)
Complement not (not in A)
Cross-Product A special set of 'ordered pairs'... more below.
Power Set Another special set- the set of all subsets!
and perhaps Symmetric Difference exclusive-or (in A or B, but not both)
Interestingly, in English we use 'or' typically in the exclusive sense, (and occasionally resort to writing 'and/or' or saying 'this or that or both'... Irritating circumlocutions) but in Latin there are two separate words that translate to the English 'or': 'aut' and 'vel'; having the two different meanings of 'exclusive or' and 'logical or' respectively.
So... To start, we have a number of rules for comparing, combinging, and creating new sets out of old ones. More in the next post, to include some sets to actually do things with!
