Re: Functions and Relations

"vc" <boston103_at_hotmail.com> wrote in news:1164116280.742477.115790_at_k70g2000cwa.googlegroups.com:

>
> NENASHI, Tegiri wrote:

>> "Aloha Kakuikanu" <aloha.kakuikanu_at_yahoo.com> wrote in
>> news:1164073723.190203.250710_at_k70g2000cwa.googlegroups.com:
>>
>> >
>> > NENASHI, Tegiri wrote:
>> >> "Aloha Kakuikanu" <aloha.kakuikanu_at_yahoo.com> wrote in
>> >> news:1164051960.913651.234880_at_m73g2000cwd.googlegroups.com:
>> >> > But then, both composition and join are
>> >> > associative. Is it merely a coincidence?
>> >>
>> >> There are relation compositions that are not associative:
>> >>
>> >> 3 -2 - 1: let const3 = 3; m2:x ->x-2; m1:x->x-1;
>> >>
>> >> 'const3 o m2 o m1'  is not associative.
>> >
>> > Relation composition or function composition?
>>
>> What is the difference ?   pretend that the composition of relations
>> is followed by a projection.
>>
>>
>> > In your example, both
>> > ways seems to produce const3.
>>
>> Please read it from left to right like lambda application ';'. I had
>> to have it written with ';'' not with 'o':  const3 ; m2; m1.  The
>> correct way: 
>>
>> m1 o m2 o const3
>>
>> But I think that you understood ;)
>>
>> >
>> > Proof by googling: "function composition associative"
>>
>> Google is wrong: now you know that not all the time ;)  The division
>> is another example,  Lie groups, et cetera.

My face is red: you are correct about the "dark side" :]. I do not know what I have thought when I wrote that the function composition is not associative. In fact, it is very simple to show that the composition is associative:

(a o (b o c))(x) => a o (b(c(x)) => a(b(c(x))) ((a o b) o c)(x) => (a o b)(c(x)) => a(b(c(x)))

Merci pour votre correction !

--
Tegi