ONT Re: Theory Of Relations
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TOR. Note 4
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In as much as relations are nothing but aggregates, sets, or logical sums
of elementary relations (or ordered tuples), we may do with relations the
whole variety of familiar set-theoretic operations that we are accustomed
to do carry out with the class of sets from which relations inherit their
properties as sets, for instance: complementation, intersection, union,
relative difference, and all the rest.
Here are a few more bits of terminology that come up at this point:
The "cardinality" |A| of a set A is, roughly speaking,
nothing more or less than the number of elements in A.
In the sorts of finite cases that presently occupy us,
roughly speaking will be ready enough.
The "power set" Pow(A) of a set A is the set of all subsets of A.
Hence, in so far as it concerns the finite case, |Pow(A)| = 2^|A|.
Let's use "bang-bang brackets", excuse my Anglish, taking the form "!...!",
as a font-shifting device, to transcribe Fraktur, Greek, or script letters,
for instance, !L! for script L, !P! for Greek Pi, !S! for Greek Sigma, etc.
If we start running out of letters, I will shift to using "scrip brackets",
taking the form "$...$", for script letters, but I'd really prefer not to.
As a convenience, let us institute the following notations:
1. !L!_1 = Pow(X^1) = {L : L c X^1} = the set of 1-adic relations on X.
2. !L!_2 = Pow(X^2) = {L : L c X^2} = the set of 2-adic relations on X.
3. !L!_3 = Pow(X^3) = {L : L c X^3} = the set of 3-adic relations on X.
As an application, let us practice the use of these conventions by
employing them to dress up the facts that we have already observed:
1. |!L!_1| = 2^(3^1) = 2^3 = 8.
2. |!L!_2| = 2^(3^2) = 2^9 = 512.
3. |!L!_3| = 2^(3^3) = 2^27 = 134217728.
To be continued ...
Jon Awbrey
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