# Negating a Statement "At least K numbers are larger than W"

What is the negation of "At least $$K$$ numbers are larger than $$W$$"?

Suppose if we set $$j=$$ number of numbers larger than $$W$$. Then, $$j \ge K$$ which negates as, $$j which translates to $$\text{at most }K\text{ numbers are larger than }W$$

Am I right?

Thanks

• You are not quite negating the way one should when doing predicate calculus, but that aside: the negation of "there are at least $K$ integers greater than $W$" over the domain of the integers is "there are at most $K-1$ integers greater than $W$". Your inequalities, while not very well defined, give this; just note "at most" is inclusive while your inequality is not inclusive, so you must use $K-1$. You could also say "at most $K$ integers greater than $W$ exclusive", though this is sure to lead to confusion. May 30, 2020 at 9:50
• Thanks, it cleared me well. May 30, 2020 at 9:54

Less than $$K$$ numbers are larger than $$W$$
Not up to $$K$$ numbers are larger than $$W$$