Post : Peirce's 1870 “Logic Of Relatives” • Comment 11.7
http://inquiryintoinquiry.com/2014/05/05/peirces-1870-logic-of-relatives-%e2%80%a2-comment-11-7/
Posted : May 5, 2014 at 6:30 pm
Author : Jon Awbrey
Peircers,
We come now to the very special cases of dyadic relations known as ''functions''. It will serve a
dual purpose on behalf of the present exposition if we take the class of functions as a source of
object examples to clarify the more abstruse concepts in Relation Theory.
☞http://intersci.ss.uci.edu/wiki/index.php/Relation_theory
To begin, let us recall the definition of a ''local flag'' L_{a @ j} of a
k-adic relation L.
• L_{a @ j} = {(x_1, …, x_j, …, x_k) ∈ L : x_j = a}.
In the case of a dyadic relation L ⊆ X_1 × X_2 = X × Y, it is possible to simplify the notation for
local flags in a couple of ways. First, it is often more convenient in the dyadic case to refer to
L_{u @ 1} and L_{v @ 2} as L_{u @ X} and L_{v @ Y}, respectively. Second, the notation may be
streamlined even further by making the following definitions:
u ⋆ L = L_{u @ X} = L_{u @ 1}
L ⋆ v = L_{v @ Y} = L_{v @ 2}
In light of these conventions, the local flags of a dyadic relation L ⊆ X × Y may be comprehended
under the following descriptions:
u ⋆ L = L_{u @ X}
= {(u, y) ∈ L}
= the ordered pairs in L that are incident with u ∈ X.
L ⋆ v = L_{v @ Y}
= {(x, v) ∈ L}
= the ordered pairs in L that are incident with v ∈ Y.
The following definitions are also useful:
u · L = proj_2 (u ⋆ L)
= {y ∈ Y : (u, y) ∈ L}
= the elements of Y that are L-related to u.
L · v = proj_1 (L ⋆ v)
= {x ∈ X : (x, v) ∈ L}
= the elements of X that are L-related to v.
A sufficient illustration is supplied by the earlier example E.
Figure 35. Dyadic Relation Example E
☞http://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-30.jpg
The local flag E_{3 @ X} of E is displayed here:
Figure 36. Local Flag Example E_{3 @ X}
☞http://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-36.jpg
The local flag E_{2 @ Y} of E is displayed here:
Figure 37. Local Flag Example E_{2 @ Y}
☞http://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-37.jpg
Regards,
Jon
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