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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