Pod pretpostavkom da krv nedostaje, davalac krvi je potreban.

Pod pretpostavkom da is a formal expression meaning assuming that, on the assumption that, or sometimes provided that depending on context.

It sounds more like written, official, or careful speech than casual conversation. In everyday Serbian, people would often use something simpler, such as:

• ako = if
• ako je tačno da = if it is true that

So this sentence is grammatical, but a bit formal in tone.

Because the preposition pod here requires the instrumental case.

The basic noun is:

• pretpostavka = assumption

Its instrumental singular is:

• pretpostavkom

So pod pretpostavkom literally means something like under the assumption.

This is a fixed, idiomatic pattern in Serbian, similar to other expressions with pod.

Here da introduces a subordinate clause: the content of the assumption.

So the structure is:

• pod pretpostavkom = under the assumption
• da krv nedostaje = that blood is lacking

English sometimes keeps that in this kind of sentence, and sometimes drops it. Serbian normally uses da.

Serbian has no articles, so there is no direct equivalent of a/an or the.

That means krv can mean:

• blood
• the blood

depending on context.

Here krv is in the nominative singular and refers to blood as a substance in general.

Because it agrees with krv, which is singular.

The verb is from nedostajati, which means to be lacking, to be missing, or to be in short supply.

So:

• krv nedostaje = blood is lacking

Since krv is singular, the verb is nedostaje and not a plural form.

Nedostajati often works a little differently from how English speakers expect.

A common pattern is:

• X nedostaje Y-u = X is lacking to Y / Y lacks X

For example:

• Bolnici nedostaje krv. = The hospital lacks blood.
• Nedostaje mi vremena. = I lack time.

In your sentence, the person or institution that lacks the blood is simply not stated. The sentence only says that blood is lacking.

Also, in real-life Serbian, many speakers would more naturally say nedostaje krvi when talking about a shortage in general.

Because after davalac the thing being given goes in the genitive case.

So:

• krv = nominative
• krvi = genitive

That gives:

• davalac krvi = blood donor

This is very similar to donor of blood in English, even though the best English translation is usually blood donor.

Davalac is the native Serbian word meaning giver or donor, depending on context.

So:

• davalac krvi = blood donor

Yes, you can also hear:

• donor krvi

and it will be understood immediately. But davalac krvi is a very standard Serbian expression, especially in official or medical contexts. A very common phrase is:

• dobrovoljni davalac krvi = voluntary blood donor

Potreban is an adjective meaning needed or necessary, and it must agree with the noun it describes.

The noun here is:

• davalac = masculine singular

So the adjective is also masculine singular:

• potreban

Compare:

• čovek je potreban = a man is needed
• pomoć je potrebna = help is needed
• ljudi su potrebni = people are needed

So potreban is used because davalac is masculine singular.

Yes, both are correct.

Serbian word order is fairly flexible, but short auxiliary forms like je tend to appear in the second position of the sentence or clause.

So:

• Davalac krvi je potreban.
• Potreban je davalac krvi.

Both are natural.

The second version, Potreban je davalac krvi, often sounds a bit more natural or flowing in everyday use. The first version puts a little more focus on davalac krvi as the topic.

It is grammatical, but it sounds somewhat formal and a little stiff.

A more natural everyday version would often be:

• Ako nedostaje krvi, potreban je davalac krvi.
• Ako nema dovoljno krvi, potreban je davalac krvi.

Why?

• ako is more everyday than pod pretpostavkom da
• nedostaje krvi often sounds more idiomatic when you mean there is a shortage of blood

So your original sentence is good for understanding grammar and formal style, but it is not the most conversational way to say it.