う・よう (Volitional)Let's, I shall (Casual volitional)

まい and the volitional form (〜よう) both express the speaker’s intent or decision, but in opposite directions. 〜よう indicates intention or willingness to do something, as in 行こう (Let’s go), while まい expresses a strong negative intention or refusal, as in 行くまい (I will not go).3

Both of these use Verb [よう], the volitional form. By itself, it expresses a suggestion, e.g. 食べよう ('Let's eat'.). Combined with としない, it means 'doesn't even try to,' e.g. 野菜を食べようとしない ('He/she doesn't even make an effort to eat vegetables.'). The key difference is between making a proposal vs. showing a lack of intent or effort.

ましょうか and Verb [よう] are both volitional forms, with ましょうか being polite and Verb [よう] (sometimes with か) casual. ～ましょうか is used to make polite suggestions like 'Shall we...?' (e.g. 行きましょうか = 'Shall we go?') 行こうか is more casual and introspective, often expressing the speaker's internal thoughts (e.g. 'Should I go?')

Both of these use Verb [よう], the volitional form. On its own, it expresses a suggestion, e.g. 食べよう ('Let's eat.'). When combined with とする, it shows an attempt to do something, often one that fails, e.g. 食べようとしたけど ('I tried to eat it, but...'). The key difference is suggestion/proposal vs. attempted action.

Both of these use Verb [よう], the volitional form, to express suggestions like 'Let's do (A).' The core meaning is the same, but they differ in tone. Verb [よう] is casual and used in everyday speech among friends. In contrast, ～ようではないか has a tone that sounds more formal and grandiose, and will be found in media or situations where the speaker is looking for dramatic effect.

Both use the volitional Verb [よう] form. On its own, it expresses a suggestion, e.g. 食べよう ('Let's eat.'). When combined with と思う, it expresses an intention or plan, e.g. 食べようと思う ('I'm thinking of eating.'). The key difference is that Verb [よう] includes others in the action, while ～ようと思う expresses the speaker's personal intent.

ましょう and Verb[よう] are both volitional forms that mean 'Let's (do something).' The difference is tone: ましょう is polite, while Verb[よう] is casual. For example, 行きましょう (polite) and 行こう (casual) both mean 'Let's go.' The casual Verb[よう] form is also used in other grammar patterns to express intent, such as 行こうと思う ('I think I'll go' / 'I'm planning to go').