ONT Re: Theory Of Relations
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TOR. Note 4
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I introduce a few more bits of
terminology that become useful
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.
Jon Awbrey
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