I have a set $A = \{a, b, c\}$ and a function that determines a value for a given element $f(x) = {value}$. Next I need the max value of all of the elements from the function:
$$a_{max} = \max_{x\in A}(f(x))$$ The problem comes with the next step where I need a set that contains all elements from $A$ without the element that produced the maximum value. So if e.g. $a_{max} = f(b)$, then I need a set $B = \{a, c\}$.
How would I write down this scenario $A, x \notin \{\operatorname{arg\,max}_x f(x)\}$ ?
Best Answer
One solution which I would consider more or less elegant is the following:
$$B = \{x \in A \mid \exists y \in A: f(y) > f(x)\}$$