to terminate / to quit
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to terminate
To formally and unilaterally end a contractual relationship, such as an employment contract, rental agreement, or subscription.
Sie hat beschlossen, ihre Wohnung zum nächstmöglichen Zeitpunkt zu kündigen.
She has decided to terminate her rental agreement at the earliest possible date.
The verb 'kündigen' (to terminate / to quit) is related to the word 'kund' (known). When you 'kündigen' something, you officially make it known that you want to end an agreement.
Er hat seinen Handyvertrag schriftlich gekündigt.
He terminated his mobile phone contract in writing.
When you end a specific agreement, 'kündigen' needs an accusative object. But when talking about leaving your job, you can also use it without an object.
Ich habe mein Zeitschriften-Abo gekündigt.
I canceled my magazine subscription.
Common things you can 'kündigen' (terminate/cancel) are 'einen Vertrag' (a contract), 'ein Abonnement' (a subscription), 'eine Mitgliedschaft' (a membership), or 'eine Versicherung' (an insurance policy).
Man kann die Mitgliedschaft im Fitnessstudio meist nur zum Jahresende kündigen.
You can usually only cancel the gym membership at the end of the year.
The idiom 'jemandem die Freundschaft kündigen' (to end one's friendship with someone) is a fixed construction. The person whose friendship is being ended is always in the dative.
Er war so enttäuscht, dass er ihr die Freundschaft kündigte.
He was so disappointed that he ended his friendship with her.
to solve / loosen
'lösen' and 'kündigen' can both mean that a contract or a fixed relationship ends. 'kündigen' is the normal word when someone formally ends an employment contract, rental agreement, or subscription. You also say 'kündigen' when you leave your job. 'lösen' sounds more general and emphasizes freeing yourself from a bond, for example from a contract or a connection. The words can be confused because both occur with contracts. In everyday language, people usually say 'den Vertrag kündigen' (to terminate the contract); 'sich aus einem Vertrag lösen' (to free oneself from a contract) sounds more formal or describes the process more generally.
Am Montag kündige ich mein teures Handy-Abo.
On Monday I’m canceling my expensive mobile phone plan.
Vor der Reise möchte sich die Gruppe aus der alten Vereinbarung lösen.
Before the trip, the group wants to free itself from the old agreement.