ONT Re: Relations And Their Divisitudes
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RATD. Discussion Note 1
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JA = Jon Awbrey
TJ = Tom Johnston
TJ: From an earlier email of yours, we have:
[begin JA:
The 1-adic "projections" Proj_X, Proj_Y, Proj_Z,
alternatively written as p_1, p_2, p_3,
as applying to a 3-adic relation L c X x Y x Z,
along with the equivalent forms of application
L_X = p_1 (L), L_Y = p_2 (L), L_Z = p_3(L),
respectively, are defined as follows:
Proj_X (L) = L_X = {x in X : <x, y, z> in L for some y in Y, z in Z},
Proj_Y (L) = L_Y = {y in Y : <x, y, z> in L for some x in X, z in Z},
Proj_Z (L) = L_Z = {z in Z : <x, y, z> in L for some x in X, y in Y}.
end JA.]
TJ: In "L c X x Y x Z", what does "L c" mean?
"Language L contains/can express"?
I'm just using "c" for the "contained in" or "subset of" symbol.
Thus, "L c X x Y x Z" just says that L is a subset of X x Y x Z.
I use the word "in" for the "element of" or membership relation.
TJ: What does "{x in X : <x, y, z> in L for some y in Y, z in Z}" mean?
Does it mean "The set of all x's in (the domain) X such that,
for each of them, there exists an <x, y, z> tuple in the
3-adic relation under discussion (in which, necessarily,
the y in that tuple is in the domain Y and
the z in that tuple is in the domain Z)"?
Exactement.
A little more colloquially,
"{x in X : <x, y, z> in L for some y in Y, z in Z}"
might be read as "the set of all x in X where
there is a triple <x, y, z> in the relation L".
TJ: And do you mean your projection to be a set, and not a multiset
(as I think another email of yours indicates you do)? If so, the
"an" in my paraphrase should be replaced by "one or more", right?
Yes^2, that would be the usual reading of the existential quantifier
signified by the phrase "for some y in Y, z in Z" in the set-builder.
Anyway, I think that's right. Let me know if it makes sense or not.
Jon Awbrey
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