"Gemüse" is singular but uncountable. It is some vegetables but not specified. Then it would be "das Gemüse".

I think the problem is that in English it's not common to say "a child eats vegetable". Usually is in the plural, "a child eats vegetables". This has to do with the countability of the word vegetable, but I'm not very good at it =/ http://www.macmillandictionary.com/dictionary/british/vegetable

There's no ein.

Same 2me :-)

Exactly the same thing happened to me too. Also reported

Later on in the tree there one along the lines of "The boy likes his entire family."

haha, that's not far from the truth!

Jajaja, took me a while.

It is the other way round. There is no plural in German. Gemüse is always singular. Das Gemüse can mean vegetable in general or a specific vegetable you are talking about. Of course you can create compound words with it. E.g. if you want to talk a certain kind of vegetable it would be die Gemüsesorte or die Gemüseart (both feminin) these words than have a plural form again i.e. die Gemüsesorten and die Gemüsearten .

I think that it is singular when there is no "das". Automatically. But try to ask someone who knows more as I am most probably much younger than you.

No, isn't genitive – there's apparently a bug that causes that skill to be filled with a lot of genitive-free sentences so let me explain.

In the written/formal-spoken standard language, the genitive is very common and hasn't declined at all, it just changed its functions. For example, in older German, genitive verbs were very common, but genitive prepositions were rare, nowadays that simply flipped over.

In colloquial German, though, the genitive case is uncommon and often avoided (mainly by replacing it with von+dative), although usage of it in everyday spoken German also depends on dialect. In Augsburg, for example, it is hardly used at all, while in Hannover, it isn't that uncommon.

Anyway, let me give a little overview of the second case here. Probably the most common manifestation of the genitive is the genitive attribute which is easy to get used to. It is equivalent to English 's and of phrases:

• die Farbe des Himmels (the colour of the sky)
• Deutschland liegt im Herzen Europas (Germany lies in Europe's heart)
• der Tod seiner Frau (his wife's death)
• die Entwicklung dieser Länder (the development of these countries)
• Claudias Buch (Claudia's book)
• der Geschmack roten Weines (the taste of red wine)

I should mention that substituting the genitive with von is obligatory in the standard language when the noun is standing alone, so one would say der Geschmack von Wein and never der Geschmack Weines. Names are an exception, where both the genitive (im Herzen Europas) and the von substitution (im Herzen von Europa) are correct.

Singular masculine and neuter nouns in the genitive use the articles des and eines, otherwise der and einer are used. The singular masculine and neuter nouns are also marked with -(e)s. A good rule of thumb is to used the -es ending for one-syllable nouns and nouns ending with a sibilant (S, Z, etc.), otherwise simply use -s. However, weak nouns will add an -(e)n instead (as is done in the dative and accusative as well). Here's a link to my explanation on weak nouns: https://www.duolingo.com/comment/10973844

All adjectives describing nouns in the genitive are declined with -en, excepting those describing feminines and plurals that aren't carried by an article, whose adjectives take -er.

The relative pronouns for masculine and neuter nouns is dessen, for feminine and plurals it's deren. To quote my memory of a Duolingo sentence: "Das ist die Frau, deren Bruder krank ist". Oddly enough, I don't remember Duolingo covering the relative pronoun for masculine and neuter nouns, but I'll just rewrite that other sentence to show it: "Das ist der Mann, dessen Bruder krank ist".

Certain prepositions use the genitive case, here are the most important (based on my research):

Big Four*:

• statt (instead of)
• trotz (despite)
• während (during)
• wegen (because of)

*These four are commonly used with dative in the colloquial language. This is unacceptable in the standard language, with the exception of "trotz" and perhaps(!) "wegen".

Location:

• außerhalb (outside)
• innerhalb (within)
• jenseits (beyond)

Common in writing:

• angesichts (in the face of)
• anhand (by means of)
• aufgrund (due to)
• ausschließlich (excluding)
• bezüglich (regarding)
• einschließlich (including)
• hinsichtlich (with respect to)
• infolge (resulting from)
• seitens (on behalf of)
• zugunsten (to the benefit of)

There are way more genitive prepositions but most of the rest are uncommon (exklusive, mangels, unweit, vorbehaltlich, zulasten, etc.). Keep in mind that, despite Duolingo's ignoring of them, not learning those 10 "written prepositions" will make reading, say a newspaper, far harder than it needs to be.

That's pretty much all you need to know since attributes and prepositions probably take up 95% of genitive usage. Learn the adjectives and verbs later; I'll just give a quick intro for each of those:

• ich bin mir meines Fehlers bewusst (I am aware of my mistake; "bewusst" causes genitive)
• wir gedenken der Toten (we remember those who died; "gedenken" causes genitive)

Finally, you can ask for the genitive with the word wessen:

• Q: wessen Hund ist das? (whose dog is that?)
• A: das ist der Hund meines Nachbarn (that's my neighbour's dog)

Viel Erfolg!