Hi adamx!
I will assume here that your comparing the elements of the sets that
have the same sub-index; that is, you're comparing a1 with b1, a2 with
b2, etc. If this is not the case, please let me know through a
clarification request. Under this assumption, the condition you are
checking could be restated as follows:
"If there is at least one a which is greater than or equal to its
'corresponding' b; and at least one b which is greater than or equal
to its 'corresponfing' a"
This condition is exactly the same as you're stating. "Not all a<b"
means that there is at least one a>=b; and the same for "Not all b<a".
As I see it, this condition can be then mathematically stated in terms
of the sub-indexes as follows:
"If there exist a pair (i,j) such that ai>=bi and aj<=bj"
The term ai, bi, etc; would be a or b with sub-index 'i'; and the same
with j. Notice also that it would not be correct to impose that the i
and j be different: if there exists one a which is equal to its
corresponding b; then the condition would be satisfied.
I hope this helps! If this is not what you were looking for, or the
assumption I made is wrong, or you have any other doubt regarding my
answer, please let me know through a clarification request. Otherwise
I await your rating and final comments.
Best wishes!
elmarto |
Request for Answer Clarification by
adamx-ga
on
14 Oct 2003 05:13 PDT
Great answer. Your interpretation of my attempt was right-on. I saw
the "backwards" approach to the relationships that you used, but
didn't think that would make stating it any clearer.
"If there exist a pair (i,j) such that ai>=bi and aj<=bj"
Is it necessary to use the term "pair" here? I ask because it muddies
the waters due to all the other pairs (e.g. a1, b1).
Could I write instead,
"If there exists (i,j) such that ai>=bi and aj<=bj"?
How would this condition be written, including punctuation, at the
beginning of the proof? Are i and j the appropriate variables to use,
or doesn't it matter? Is it okay to use i, j even if my set of numbers
in the statement uses n instead? E.g. a1, b1, a2, b2, . . . an, bn
Thanks!
|
Clarification of Answer by
elmarto-ga
on
14 Oct 2003 07:49 PDT
Hi adamx!
If you are already mentioning "pairs" when referring to a1,b1, etc,
then you can perfectly drop the term "pair", and it will still be
right. Moreover, if the notation (i,j) (with the parenthesis) bothers
you, perhaps because you are already using (a1,b1), you can also drop
the parenthesis:
"If there exist i, j such that ai>=bi and aj<=bj"
Another option is, if the set a1, a2,... is called A:
"If there exist ai, aj belonging to A such that ai>=bi and aj<=bj"
I don't think it's confusing to use the i and j. These are very
standard: most of the time, i and j are used either for complex
numbers or for indices. Also, the n shouldn't matter because you're
talking about different things: a1, a2,...,an means that the *last*
element has sub-index n; while i and j refer to any sub-index.
The punctuation should be right as it is. However, if you don't use
the parenthesis, you might want to use a different type for i, j (such
as italics), so that this comma doesn't get mixed with a "real" comma.
Please let me know what you think of this.
Best wishes!
elmarto
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