Ég þarf að segja upp leigusamningnum.

Þarf is the 1st person singular / 3rd person singular present tense form of þurfa, which means to need or to have to.

So:

• ég þarf = I need
• hann/hún/það þarf = he/she/it needs

In this sentence, Ég þarf að... means I need to... or sometimes I have to..., depending on context.

Here, að is the infinitive marker, like English to in to say, to cancel, or to terminate.

So:

• segja = say
• að segja = to say

After þurfa, Icelandic normally uses að + infinitive:

• Ég þarf að fara = I need to go
• Ég þarf að lesa = I need to read
• Ég þarf að segja upp... = I need to cancel/terminate...

So að is not the preposition to here; it is the marker that introduces the infinitive.

Yes, segja by itself usually means to say or to tell. But segja upp is a set expression / phrasal verb, and its meaning changes.

Depending on context, segja upp can mean:

• to terminate
• to cancel
• to give notice on
• to fire / dismiss
• to quit / resign in some contexts

So you should understand segja upp as a whole unit, not by translating segja and upp separately.

In this sentence, with leigusamningnum, it means something like to terminate the lease or to give notice on the rental agreement.

Literally, upp does mean up, but in segja upp it is part of the verb expression and does not have to be translated literally.

This is similar to English phrasal verbs:

• give up
• call off
• break up
• set up

You usually cannot guess the full meaning just from the individual parts. In the same way, segja upp has to be learned as a vocabulary item.

Because segja upp takes the dative case for the thing being terminated.

The dictionary form is:

• leigusamningur = lease agreement / rental contract

But in this sentence, it appears in the dative singular definite form:

• leigusamningnum = the lease agreement (in dative)

Very roughly:

• leigusamningur = a lease agreement
• leigusamninginn = the lease agreement (accusative)
• leigusamningnum = the lease agreement (dative)

The important point is: this form is chosen because the verb expression segja upp governs the dative.

The ending -num is a strong clue that this noun is:

• singular
• definite = the
• dative

So leigusamningnum means the lease agreement in the dative case.

You can break it down like this:

• leigusamning- = the noun stem
• -num = dative singular definite ending

This is one of the forms Icelandic learners often have to memorize along with the noun’s gender and declension pattern.

The base form is leigusamningur.

It is a compound noun made from:

• leiga = rent, rental
• samningur = agreement, contract

So leigusamningur literally means rental agreement or lease contract.

Compounds are extremely common in Icelandic, and this is a very typical example.

Yes, the position of upp can vary depending on the structure.

In an infinitive phrase like this one, it commonly stays next to the infinitive:

• að segja upp leigusamningnum

But with a finite verb in a main clause, you may often see the particle separated:

• Ég segi leigusamningnum upp.

So Icelandic can behave a bit like English phrasal verbs, where particles may shift position. The exact placement depends on the grammatical structure.

For this sentence, að segja upp leigusamningnum is completely normal.

Usually þurfa að means need to or have to. It expresses necessity, but not always the same kind of strong obligation as English must.

So depending on context, it could be understood as:

• I need to terminate the lease
• I have to terminate the lease

If you wanted a very strong must, Icelandic can also use verða að in many situations:

• Ég verð að segja upp leigusamningnum = I have to / I must terminate the lease

But in everyday speech, þurfa að is extremely common and natural.

Yes. Segja upp is used in several related contexts, and the exact meaning depends on the object.

For example:

• segja upp starfsmanni = dismiss an employee
• segja upp vinnunni = quit the job
• segja upp samningi = terminate a contract
• segja upp leigusamningi = terminate a lease

So the core idea is often ending some formal relationship or arrangement, but the English translation changes with context.

For most learners, the safest approach is to memorize it with the case it governs.

So it is useful to learn it as:

• segja upp + dative

That way, instead of memorizing just the meaning, you memorize the whole pattern:

• segja upp samningi
• segja upp leigusamningi
• segja upp starfi or other dative forms depending on context

Icelandic has many verbs and verb phrases that require a specific case, and this is one of them. Over time, these patterns become more natural, but at first they usually need to be learned individually.