عضوية المكتبة مهمة.

Because this is a nominal sentence in Arabic. In the present tense, Arabic usually does not use a separate word for is/are.

So:

• عضوية المكتبة = the subject/topic
• مهمة = the predicate/comment

The full idea is simply Library membership is important or The library membership is important, even though Arabic does not say is out loud here.

In grammar terms:

• عضوية المكتبة = مبتدأ
• مهمة = خبر

Yes. عضوية المكتبة is an iḍāfa (construct phrase).

An iḍāfa is a noun + noun structure that often corresponds to English expressions like:

• the membership of the library
• library membership

Here:

• عضوية = membership
• المكتبة = the library

So together, عضوية المكتبة means something like the library’s membership or library membership, depending on context.

Because in an iḍāfa, the first noun normally does not take ال.

So Arabic says:

• عضوية المكتبة

not:

• العضوية المكتبة

Even though عضوية does not have الـ, the whole phrase can still be definite because the second noun, المكتبة, is definite.

That is a key rule of iḍāfa:

• the first noun does not take الـ
• the second noun shows whether the whole phrase is definite or indefinite

Because the second term of an iḍāfa is put in the genitive case.

In fully vowelled Arabic, the sentence is:

عُضْوِيَّةُ الْمَكْتَبَةِ مُهِمَّةٌ

Notice:

• عضويةُ ends with ـُ because it is the subject
• المكتبةِ ends with ـِ because it is the second noun in the iḍāfa
• مهمةٌ ends with ـٌ because it is an indefinite predicate

In normal writing, these short vowels are usually not written, but the grammar is still there.

Because it matches عضوية, which is a feminine singular noun.

In Arabic, the predicate adjective usually agrees with the noun in:

• gender
• number

So:

• عضوية is feminine singular
• therefore مهمة is also feminine singular

If the noun were masculine, you would use مهم instead.

Because here مهمة is the predicate of the sentence, not an adjective directly modifying the noun.

Arabic often uses an indefinite predicate in sentences like this:

• عضوية المكتبة مهمة = Library membership is important

If you said:

• عضوية المكتبة المهمة

that would usually mean the important library membership, which is a noun phrase, not a complete sentence.

So the lack of الـ on مهمة helps show that it is the predicate: important.

A useful pronunciation is:

ʿuḍwiyya

But in this sentence, because the word is connected to another word in an iḍāfa, the final ة is pronounced like t in full reading:

ʿuḍwiyyatu l-maktabati muhimmatun

or without full case endings:

ʿuḍwiyyat al-maktaba muhimma

So the important point is:

• ة at the end of a word often sounds like -a / -ah when pausing
• but it sounds like t when connected grammatically, as in an iḍāfa

That is why عضوية المكتبة is pronounced more like ʿuḍwiyyat al-maktaba than ʿuḍwiyya al-maktaba in careful reading.

In fully vocalized MSA, it is:

عُضْوِيَّةُ الْمَكْتَبَةِ مُهِمَّةٌ

Breakdown:

• عُضْوِيَّةُ = -u because it is the subject of the nominal sentence
• الْمَكْتَبَةِ = -i because it is the second term of the iḍāfa
• مُهِمَّةٌ = -un because it is an indefinite predicate

In everyday reading and speech, these endings are often reduced or dropped, especially at pause.

It is functioning as the predicate of the sentence, not as a regular attributive adjective inside the noun phrase.

That distinction matters:

• عضوية المكتبة مهمة = The library membership is important
• العضوية المهمة or عضوية المكتبة المهمة = an adjective modifying a noun, like the important membership or the important library membership

So in this sentence, مهمة tells us something about عضوية المكتبة. It is not just attached to it as a descriptive adjective within the same noun phrase.

It is singular.

• عضو = member
• أعضاء = members
• عضوية = membership

So عضوية is an abstract noun meaning membership, not a plural noun meaning members.

You could, but it is usually not necessary in a simple present-tense sentence like this.

The most natural basic MSA form is:

عضوية المكتبة مهمة

Adding هي can give extra emphasis or be used in certain styles, but beginners should learn that Arabic normally does not need a separate present-tense is.

So the simple sentence without هي is the standard form.