Inquiry Blog
http://inquiryintoinquiry.com/2015/12/08/relations-their-relatives-16/
http://inquiryintoinquiry.com/2015/12/10/relations-their-relatives-17/
http://inquiryintoinquiry.com/2015/12/12/relations-their-relatives-18/
http://inquiryintoinquiry.com/2015/12/22/relations-their-relatives-19/
Helmut, List,
I used braces {...} to indicate a set {x, y} of two elements.
Written order makes no difference to sets, so {x, y} = {y, x}.
I used parentheses (...) to indicate an ordered pair (x, y).
The ordered pairs (x, y) and (y, x) are distinct if x ≠ y.
We say that a dyadic relation L is "symmetric"
if (x, y) being in L implies that (y, x) is in L.
Regards,
Jon
On 12/23/2015 11:54 AM, Helmut Raulien wrote:
Supplement: I think, in my below text there are a lot of mistakes: I mixed up
tuples with products, I am not sure when to use round and when winged brackets,
and the set of unordered triples does not consist of three, but of six sets of
ordered ones, and I dont know what else. I must read your wikipedia papers about
relations first, I think. It is very interesting, eg. it seems to me, that other
than with dyadic products, with triadic ones there are a lot of different kinds
of symmetry, rotational and linear. Happy Christmas and happy new Year!
Jon, list,
about ordered and unordered pairs: In the mathematical books I had read in,
there was only the way of writing ordered pairs. And symmetry was only explained
by the example of a subset of a product of two same sets (A x A). I had thought
then, if you have two different sets, A and B, symmetry makes no sense, lest you
look at a subset of {(A x B) U (B x A)}. You wrote, the short way of writing
this, is {A x B}. That would be an aggregate of unordered pairs. Now, if you
have three sets, A, B, C, then how do you write the unordered cartesian product?
Should be {A, B, C} written in a triangle, with one "x" in the centre, or "{A x
B x C}"? (short form) ? Or {(A x B x C) U (B x C x A) U (C x A x B)} (long
form)?
I still find relation reduction interesting. I must read your paper again, about
the projective reduction, and try to find out, whether it is possible to
projectively reduce a triadic relation of R, O, I to three dyadic ones, but not
R-O, O-I, I-R, but R-R, R-O, R-I. Because I think, this is the Peircean way.
Have you tried that? Im not sure, when I will have time resp. overcome my
laziness to, well, just take a piece of paper and start. To just start is always
the most difficult part, like with the tax declaration, but other than with
that, there is no penalty if I dont.
Best regards,
Helmut
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