make sure to be absolutely clear about the difference between a sufficient and a necessary condition from “A Short Lesson”, and the ways that can be used to express those conditions (e.g., “if…then…” vs. “only if…”, etc.)
Please do read “X implies Y” as saying “X is a sufficient condition for Y”. I’ve included a clarificatory note. Big hint: Please revise what it means for something to be a necessary condition for something else. Another big hint–this is actually meant to be a simpler question than many of you seem to be taking it. Everything you need is already in the question itself!
Small clarificatory note added for Dave’s statement. When we say that “given P, Q, R, it can still be X”, we are basically saying that {P, Q, R} does not rule out X.