"Нет, у него нет брата."
Translation:No, he does not have a brother.
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The English provided here is sound. Are you saying the English translation of the Russian sentence is incorrect, or are you saying that your translation is the preferred one? Because I'd have to respond with 'wrong' on both counts.
Whereas it is uncommon in most countries today not to have more than one wife, it is definitely not unheard of. 'He has no wives' is an equally valid statement.
Similarly, if someone asks 'Wasn't Jimmy Tom's brother?' one might respond with either 'He has no brothers' as well as 'He has no brother'. Both are correct and would not make anyone confused about the intended meaning.
I don't really understand how Genitive case is connected to "-ing". :o I think they're completely different things.
Well, as for cases, basically Russian nouns have several form. For example, nominative case is used for subject of the sentence and for both nouns in "X is Y"-type sentences. Most other cases have several possible uses.
The most obvious use of genitive is indicating possession: if you put it after any noun, it will mean a possessor. E.g. in «дом Мари́ны» 'Marina's house', Мари́ны is Genitive.
Other use of genitive case is indicating absence. You use it after «нет», and sometimes, when the verb is negated, its object is in genitive (не зна́ть сло́в 'not to know the words'; сло́в is Genitive).
«Нет» always requires genitive: «нет воды» 'there is no water', «нет бра́тьев» 'there are no brothers', «нет ве́тра» 'there is no wind'.
Another prominent use of genitive is indicating possession: «температу́ра воды́» 'temperature of the water', «исто́рия бра́тьев» 'the brothers' story', «направле́ние ве́тра» 'direction of the wind, wind direction'.
Genitive is often used in negative sentences, when denoting an object: «Я не пила́ воды́» 'I didn't drink water', However, it's never used when talking about living people. Accusative is also used in this function, and accusative is more common. Genitive works best for abstract things and uncountable nouns.
Sometimes genitive is used for a direct object in positive statements and questions. Then, it has the meaning 'some, part of': Я вы́пила воды́ 'I drank some water'. Accusative is more common for direct objects: Я вы́пила во́ду. I've drank the water.
Some prepositions require genitive: у воды́ 'near the water', у бра́тьев 'at the brothers' possession, at the brothers' place', из воды́ 'out of the water', от бра́тьев 'from the brothers'.
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Why do we use "нет" for both "no" and "not" here? I was under the impression that this never happens.