>> I am pairing values n with n'.
>> Let i_1 and i_2 be integers (whether distinct or not) such that 1<=i_1
&
>> i_2 <= y.
>
> Did you mean x <= i_1, i_2 <= y? Because if you really
> meant the lower bound to be 1, x doesn't figure in your
> definitions of pairs (n, n') and therefore shouldn't be
> involved in any expression counting such pairs.
No, sorry; I meant in fact that 0 <= i_1, i_2 <= y-x+1.
Then I go on to say, n=y-i_1 and n'=y'-i_2
>> There are (y-x)^2 possible pairs n and n'.
>
> This makes no sense until you've told us what n and n' are.
n is an integer in the interval [x,y] and n' is an integer in the inteval
[x',y'].
>
>> This includes pairs
>> (n,n') such that n=y-i_1 and n'=y'-i_2, and also (n,n') such that
n=y-i_2
>> and n'=y-i_1.
>
> Do you mean that for each i_1 and i_2 in the range specified
> above, (y - i_1, y' - i_2)
Yes.
>> Let C be the set of the possible combinations of (n,n').
>> #C=(y-x)^2.
>
> If my interpretation is correct, #C is actually
> (y - x + 1)^2, since there are y - x + 1 integers in the
> range [x, y] (assuming that x and y are themselves integers
> and x <= y).
Quite right.
>> Define A' to be a subset of C, such that
>
>> a) #A' = y-x
>
>> b) y>x.
>
>> c) y'-x'=y-x
>
> This makes no sense: what are x' and y'?
x' is the lower bound and y' the upper bound of the interval [x',y'] .
With thanks.
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