I want to say "for all k from 1, 2,...B that are not elements of the vector S. What is the best/most professional way to represent this? Is there a better way to represent it than
Correct would be something like
$1 \le k \le B$, $k \notin S$. Elipses are ambiguous...
Don't try to reduce everything to symbols, that easily turns into utter gibberish. What you write is for humans to understand, symbols (particularly less familiar ones) just stand in the way.