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 Duality - is it necessary ?

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Posts : 95
Join date : 2009-09-15
Age : 55
Location : Alexandria

PostSubject: Duality - is it necessary ?   Sat Oct 17, 2009 4:53 pm

One way or another the duality occurs in math.

For example, let me define the dual of a pair (a, b) as (b, a) with the notation
(a, b)* := (b, a) Of course,
(a, b)** = (b, a)* = (a, b). But this duality is more touchy : for example, we can prove diff( (f, g)*) = (diff(f, g)) * and so on.

Another example of duality : Let A = { 1,2,3, 4 } be a set of individuals x and f, g, h one-to-one maps on A onto K = {a, b, c, d,}

- | f, g, h
1 | a b c
2 | b c a
3 | c d b
4 | d a d

We have f(1)= a, f(2)= b,..., h(4) = d.

There is a risk that someone that does not know the 'right' notations to see the table as functions defined on { f, g, h } :

1(f) = a, 1(g) = b,... 4(h) = d; so, x(f) = f(x).
Thus, to maintain the coherence of 'right' notations, we write x*(F) instead of x(F) to say the same thing. Of course, x**(F) = (x*(F))* = (F(x))*= F*(x) = x(F) for every F and every x.

Now comes my question : Has anyone an example where the duality brings more than a rewriting of something, when it is necessary to a result that can't be proved else ?

The first example could be the contrapositive :
contrapositive (contrapositive (implication)) = implication
To prove that sqrt(2) is not rational, one needs to prove the contrapositive.
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