define the set of floating point numbers by $F(2,53,-1022,1023)$.
A Sum $(a+b)$ with $a,b \in F$ is said to be good if $(a+b)\in F$ otherwise it is a bad sum. I have to figure out if there are more bad or good sums.
My attempt : Since $F$ is finite it has a maximum element call it $M$. Now i fix $M$ and form the sums of $x+M$ for all $x\in F$. Each of this sum will not be in the set because it is bigger then it's maximum value. Also there are $n$ such sums (all different from each other) , where $n$ is the number of elements in $F$. So the number of bad sums is at least $n$ , while the number of good sums can be at most $n$.
This is an old exam question and the solution to the problem is totally different to mine ( it depends on the parameters of the set while my argument is general) so i would like to know if my reasoning is ok or if not , where does it fail? Thank you for the feedback!
Ps: If you have troubles with understanding the notation please let me know and i will try to improve the question , thanks.
I don't understand your notation, but your proof contains the claim that "the number of good sums can be at most $n$", which is incorrect. If $F$ were the interval $[1,100]$, then indeed there are $99$ sums $1+100, 2+100,\ldots, 99+100$ all of which are bad. However there are ${50\choose 2}=1225$ sums of two distinct numbers each in $[1,50]$, which are all good.