We are walking the line between what is technically correct and what people might say. Technically putting "neither" before a verb in a case like this means that you are negating two verbs (I neither have nor want a cat), not two objects (I have neither a cat nor a dog). Technically then the meaning of "I neither have..." is different than the Russian sentence. This is a tricky case because while people do use the order you are suggesting, it's not nearly as common as something like "Who are you going with?" where it's obvious we should accept both "whom" and "who."
After «не́т» we use genitive case; and «ни... ни...» doesn't affect the cases. «Ко́шки» and «соба́ки» are genitive singular, while «ко́шек» and «соба́к» are genitive plural. So, if you wanted to say it in plural, you’d use:
- У меня́ нет ни ко́шек, ни соба́к. 'I have neither cats nor dogs.' (=I have no cats, and no dogs.)
However, this sentence uses singular nouns. They may mean the same thing (well, if you don’t have a single cat, it also means you don't have cats), but this course expects you to use the closest forms grammatically to make sure you understand the grammar.
Because «нет» requires genitive, not nominative.
I'm not sure if you've learnt the difference, but in short: Russian nouns have several forms called case forms. The most basic one is 'nominative' case. You use it for the subject of the sentence, and this is the form listed in the dictionaries. Nominative singular forms are ко́шка, соба́ка. Nominative plural forms are ко́шки, соба́ки.
But to express absence, you use «нет» with a different case form, genitive. Genitive singular is ко́шки, соба́ки. Genitive plural is ко́шек, соба́к. Yes, genitive singular often concedes with nominative plural. You'd need to distinguish them by context (for example, in negative sentences with «нет» you expect genitive and not nominative).
By the way, со́шка is not the same as ко́шка! Со́шка means 'someone small and unimportant', and it's normally used together with an andjective «ме́лкая»: «ме́лкая со́шка» 'someone unimportant'.
Russian nouns have several forms called cases. The cases show the role performed by the person or thing described by the word.
For example, in 'I help her', I is the nominative case, it means 'I' is the one who does the action of helping. In 'She help me', 'me' is the objective case, it means 'I' am affected by someone else's action of helping. So, the form shows who is doing the action and who is affected by it.
The Russian takes the concept further: all nouns (not just 'I' and 'she', but also 'cats', 'dogs', etc.) change their forms in the sentence. Also, Russian has 6 cases.
You can see all forms of the word ко́шка in the Wiktionary: https://en.wiktionary.org/wiki/%D0%BA%D0%BE%D1%88%D0%BA%D0%B0#Declension_2 (click on the 'Declension of ко́шка' box near the end of the page if it's hidden). This table might look overwhelming, but don't worry: this course will present all the cases one-by-one, in bite-sized lessons.
In this sentence, the difference between nominative and genitive is important.
Nominative case is used for the subject of the sentence, someone doing the action. Nominative case is also used in the sentences 'X is Y' for both X and Y. It is also used in positive 'there is'-type sentences that state existence:
- Ко́шка охо́тится. 'A/the cat hunts.' (ко́шка is the subject of the sentence)
- Ко́шки охо́тятся. 'Cats hunt.' (ко́шки are the subject of the sentence)
- Ко́шка — э́то живо́тное. 'A cat is an animal'. ('X is Y'-type sentence, both ко́шка and живо́тное are in the nominative-case form)
- Ко́шки — э́то живо́тные 'cats are animals' ('X is Y'-type sentence, both ко́шки and живо́тные are in the nominative-case form)
- В до́ме есть ко́шка. 'There is a cat in the house.' (positive 'there is'-type sentence, ко́шка is nominative)
- В до́ме есть ко́шки. 'There are cats in the house.' ('there is'-type sentence, ко́шки is nominative)
- В до́ме есть ко́шка. 'There is a cat in the house.' ('there is'-type sentence, ко́шка is nominative)
- У меня́ есть ко́шка. 'I have a cat.' (literally: 'At my [possession], there is a cat'; 'there is'-type sentence, ко́шки is nominative)
- У меня́ есть ко́шки. 'There are cats in the house.' (literally: 'At my [possession], there are cat'; 'there is'-type sentence, ко́шки is nominative)
Genitive case is another form. It has a number of uses. For example, it is used in 'X of Y' construction for Y (and 'of' is left untranslated). It is used with «нет» to express the meaning 'there is no':
- еда ко́шки 'cat's food, food of a/the cat',
- еда ко́шек 'cat's food, food of [the] cats',
- В до́ме нет ко́шки. 'There is no cat in the house.' (нет is used to express 'there is no', so a genitive is required: ко́шки is genitive)
- В до́ме нет ко́шек. 'There are no cats in the house.' (нет is used to express 'there are no', so a genitive is required: ко́шек is genitive)
- У меня́ нет ко́шки. 'I have no cat.' (literally: 'At my [possession], there is no cat'; нет is used to express 'there is no', so a genitive is required: ко́шки is genitive)
- У меня́ нет ко́шек. 'I have no cats.' (literally: 'At my [possession], there are no cats'; нет is used to express 'there are no', so a genitive is required: ко́шек is genitive)
As you can see, ко́шки can be both nominative plural and genitive singular. (The same is true for «соба́ки».) But «нет» requires genitive, so we know that it's genitive singular.
So, you should use singular forms ('a cat' and 'a dog') in the translation, because the Russian sentence uses the singular forms.
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Russian word for 'parrot' is «попуга́й».
Interestingly, попуга́й is also an imperative of попуга́ть 'to frighten, to scare (a bit, for some time)'. There is a funny children’s poem that uses this similarity:
Говори́т попуга́й попуга́ю: 'A parrot says to a parrot:'
«Попуга́й, я тебя́ попуга́ю!» 'Parrot, I will frighten you [a bit]!'
Попуга́ю в отве́т попуга́й: 'The parrot [says] to the parrot in answer:'
«Попуга́й, попуга́й, попуга́й!» 'Frighten me [a bit], frighten me, frighten me!'
Words marked in italics are forms of the verb попуга́ть 'to frighten', while other words are form of the noun попуга́й. Note that those words are completely unrelated: попуга́й as a bird is a loanword, while попуга́ть 'to frighten [a bit]' is created from the Russian word пуга́ть 'to frighten'.
In the previous question, it says 'ни рис ни яблоки', and the translation is 'rice' and 'apples', so I suppose we should use nomative after 'neither'. However, the translation provided for this sentence is 'neither A cat nor A dog', then the cat and the dog will not be nomative. So what are we supposed to use after ни actually?
English is my native language, I was typing from mobile, those were typos. But that still does not address the point that "do not" and "don't" should both be accepted as correct as they are essentially the same, as"don't" flows better. So instead of fixing English comments, maybe try providing an explanation on why the translation would not work with "don't".