- I let him drink my milk - Я позволила ему выпить моё молоко
This is the main meaning of "let", but there are tons of other meanings in my dictionary %) I was shocked by "set", "let", "take", "put" when I learned English at school, but with time I got to understand them "the English way", i.e. without translating into Russian word by word and struggling with many meanings.
Russian verbs come in two flavours: perfective and imperfective.
We use perfective verbs to express a single action (such use is associated with a starting point, a result, or another way of limiting the action's span). We use imperfective verbs to express an ongoing process, a habitual or repeated action. Only those imperfective verbs can be used in the present tense.
Adding a prefix is the most popular way of making perfective verbs from an imperfective base (e.g., писа́ть → написа́ть, записа́ть, вы́писать, расписа́ть, переписа́ть ...) Oftentimes a "neutral" perfective exists than means basically the same action as the original verb, only converted to a result or an outset.
Anyway, the use of Давай/Давайте depends on which verb you use. If you use an imperfective verb (to suggest an immediate process or a habitual action), it is the infinitive that you use (e.g., Давайте готовить что-то). If you use a perfective verb, the verb should be in the 2nd person plural (e.g., Давайте приготовим что-то).
Good question, it's just because pust' is a bit of a strange exception here compared to alternatives like позольте/разрешите (these would use ему). User Olimo described it as a "grammatical particle" that doesn't change form, so... just one of those things to remember I suppose.
Here more examples: http://masterrussian.com/vocabulary/pust_let.htm
And also "...the word пусть, which translates as "let" and used in the same way it is in English. For example: "Пусть они едят торты," ("Let them eat cake.") Note how the verb is conjugated in agreement with the subject, and not changing due to пусть." (http://ielanguages.com/russian2.html)
Then maybe "Let him make dinner" is a bad translation, as it means the same as "Allow him to make dinner", which is simply more formal. This is not a suggestion, but means "He wants to make dinner, so don't stop him. "Let's make dinner" is however a suggestion. (I know, it's strange!) A suggestion would be " How about him making dinner?"
Yes. "Him", like any other pronoun and 99.9% of nouns, will take various forms depending on what case the pronoun "he" is in. In accusative then его (Я вижу его); in dative then ему (Я даю ему деньги); in genitive then (н)его (Мы делаем это без него); in instrumental then (н)им (Мы делаем это с ним); in prepositional then нём [though this would presumably be in reference to a masculine building or facility, though I guess a doctor might diagnose an illness "in him"] (в нём).
I am pretty sure the english usage is anachronistic. 'Let' is quite formal in English for the 3rd person singular, I feel, deriving from literature and ceremony. 'He can cook dinner', in context, carries the suggestion/permission meaning today, and is a worthy translation IMO.
It means you do not care about some circumstances or someone's actions you've just been notified about. Let things be as they may, you don't give a damn.
Also «Ну и ладно». An important difference is that when saying Ну и пусть/Ну и пускай, you can add a personal verb form to state what you do not care about (after all, this is what пусть normally does). Ладно cannot attach any verb forms.
Note also the concessive use of пусть, which makes it even easier to understand:
- Пусть ты и не писатель, но пишешь ты очень красиво. ~ Even though your are not a writer, your writing is very beautiful.
They are not. This is the 3rd person imperative, which is not what you usually think of as an imperative. Пусть is a fixed form.
Russian 2nd imperatives end in -и(те), -й(те) or -ь(те), with -те added for polite "you". You can build it if you know the non-past stem:
- дать "give" → дай/дайте
- читать "read" → читай/читайте
- говорить "speak" → говори/говорите
- спать "sleep" → спи/спите
- писать "write" → пиши/пишите
- есть "eat" → ешь/ешьте (irregular)
- идти "go" → иди/идите
- плакать "cry" → плачь/плачьте
- ставить "place, put" → ставь/ставьте