won't; hasn't; isn't; doesn't
Polite negative question used to ask whether something is not the case. Often expects confirmation or expresses surprise.
See also: ます
Doesn't this train go to Tokyo?
won't you; shall we; why don't we
Polite invitation, suggestion, or indirect command. Softens a request by framing it as a negative question.
Haven't you finished your homework yet?
Won't you go see a movie with me?
Shall we get going soon?
ませんか is the polite form used in formal speech, while ないか is the casual equivalent used among friends or family.
Polite negative form of the auxiliary verb ます (masu) combined with the question particle か (ka).