Translation:a cat and the cat
I think in this exercise, Duolingo is trying to show you the difference and comparison between definite and indefinite nouns in Arabic and English.
In English, typically, an indefinite noun comes after "a/an" - Rarely though in some general sense the word is mentioned without "a/an", like for example I love nature (the word nature here comes without any indefinite article because it is a general word for a general thing that cannot be counted).
Regardless, "cat and the cat" should be accepted since there is no context to dictate that it has to be interpreted as an indefinite.
If we are translating from Arabic, then Tanwin can be considered a sign for the indefinite (hence the two cannot appear in a single word). Besides, does cat and the cat bear any sense or logic without the indefinite article a in this sentence specifically? Not to me.
Well, "a cat and the cat" doesn't bear any sense or logic to anybody on the planet either! So since this isn't even a sentence, a person who reads the word "cat" and translates it "cat" without grammatical particles shouldn't be penalized. Why lose a heart in the exercises when you've shown you understand?
And no, tanwin is not a sign of the indefinite.
محمدٌ رسول الله
refers to a specific Muhammad. Tanwin can't occur with the definite article, that's true, but that in itself does not mean that it indicates indefiniteness.
you are mixing principles. I know محمد is a proper noun and it gets Tanwin. Also Tanwin is a marker for adverbs حال and a number of cases. I'm saying in one class of words, it can be considered a sign of an indefinite word, noun or adjective.
The logic I'm talking about is the linguistic logic. Mentioning "cat" without any article is categorized as a general term. Like saying Man was put on the moon, where "Man" is a general term referring to the humankind, not a specific man.
The sentence above bears a contrast between definition and indefinition of a word and naturally when translating from Arabic one would add "a/an" to the indefinite noun or adjective, unless the term is meant to be general.
And I'm saying tanwin is never a sign of indefiniteness. It happens to occur with indefinite words sometimes, but you can't call it a sign if you yourself admit that some definite words get it and some indefinite words don't get it. What does "sign" even mean then?
Again, you say "the sentence above," when what is above is not a sentence; it's a phrase, and this matters. قطة here is a word, simply an utterance. It performs no grammatical role, at all. It is void of any meaning, beyond that of "cat." If there is a song that says "I love this cat, cat, beautiful cat," you can translate it "2u7ibbu haadhihi l-qittah, qittah, qittatun jamiila," or "al-qittata, qittata, qittata l-jamiila," or "al-qittata, qittatun, qittatun jamiila," because without context, you could interpret the following "cat"s as an echo of the previous sentence, or an introduction of a new one.
You don't need to agree with me, but the mere fact that people who think like me exist means that this should be accepted as an answer by Duolingo, since this utterance doesn't have any semantic value and is translated solely as a string of items. I'm going to report it anyway, and it's up to the moderators to make the final decision, I guess.