On Wed, 4 Feb 2004 06:18:30 +0000 (UTC), "John Edser"
 wrote:


>JM:-
>...C' contains only two things, not four.  Those two things are both
>sets, and each of them contains two of the original objects.  The set
>"container" does not "dissolve" when a set
"moves inside" another set.
>
>JE:-
>The set container inside, does dissolve when sets are 
>joined by set union. This is a critical difference
>between union and intersection. AW severely criticised me when 
>I suggested that sets merged by set union lose their integrity 
>because of this fact. Set union is a non reversible logic
>but set intersection is a reversible logic. Understanding
>this difference is critical for any application of set theory 
>to evolutionary theory. If independent sets of fitness are merged
>by set union they lose their fitness independence but this
>is 100% preserved when fitness sets intersect.
>
>Can I make a suggestion?
>
>Could AW and/or JM please supply a valid example
>of any applied set intersection where all of one set 
>is a subset of the other, i.e. provide a simple problem
>and then show how set theory correctly illustrates this problem 
>where  the illustration shows one set as a subset of the other.
>Could you please point out why you think the intersection
>was mathematically valid in the example provided.
>

I find it extremely difficult to understand just why John Edser has so
much trouble with elementary set theory.  There is no big conceptual
difference between set union and set intersection as he claims.  There
is no such thing as a "set container".

Consider the sets  S1 = { a, b, c } and S2 = { a, b, d }. 

Then the intersection of S1 and S2 is simply the set { a, b } and the
union of S1 and S2 is the set {a, b, c, d }.  Taking the union or
taking the intersection do absolutely nothing to the original sets.
They still exist unchanged. The intersection and the union are brand
new sets, not modifications of the original.

Now consider the case requested where one set is a proper subset of
the other:  S1 = {a, b, c } and S3 = {a, b}.  Now the intersection of
S1 and S3 is {a, b} which is equal to S3 while the union of S1 and S3
is {a, b, c} which is equal to S1.  The fact that the union and the
intersection happen to be equal to (contain the same elements) as one
of the originals has absolutely no signficance on whether the original
sets are somehow modified -- they are not.  

What is the big deal?  Why has there been a thread about set
intersection for all these many months?
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