
aussetzen
to abandon
To leave behind someone or an animal that depends on help, and no longer take care of them.
Es ist herzlos und verboten, Haustiere einfach im Wald auszusetzen.
It is heartless and illegal to simply abandon pets in the forest.
to abandon / suspend / expose
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to abandon
To leave behind someone or an animal that depends on help, and no longer take care of them.
Es ist herzlos und verboten, Haustiere einfach im Wald auszusetzen.
It is heartless and illegal to simply abandon pets in the forest.

to sit out
To interrupt an activity, participation, or a round in a game, either voluntarily or because one has to.
Mir ist schwindelig, ich glaube, ich muss bei der nächsten Runde aussetzen.
I feel dizzy; I think I need to sit out the next round.

to find fault with
To criticize someone or something; to find faults or shortcomings.
An seiner Arbeit gibt es wirklich nichts auszusetzen, sie ist perfekt.
There really is nothing to find fault with in his work; it's perfect.

to expose
To expose someone or something to a particular condition, danger, or influence.
Man sollte seine Haut nicht zu lange der direkten Mittagssonne aussetzen.
You shouldn't expose your skin to the direct midday sun for too long.

to suspend
To temporarily put something out of force or postpone it, such as a punishment or regulation.
Das Gericht hat die Haftstrafe zur Bewährung ausgesetzt.
The court suspended the prison sentence on probation.

to offer
To offer a reward or prize for a particular achievement or piece of information.
Für Hinweise, die zum Täter führen, wurde eine Belohnung von 5000 Euro ausgesetzt.
A reward of 5,000 euros was offered for information leading to the offender.
'Aussetzen' is a separable verb. The prefix 'aus' comes at the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Er setzte eine hohe Belohnung aus.
He offered a large reward.
The verb consists of the prefix 'aus-' (out) and 'setzen' (to set, to place). The basic idea of 'putting something outside' or 'leaving something outside' helps with understanding many meanings, e.g. leaving an animal outside or staying out of a round.
Wer sein Tier aussetzt, macht sich strafbar.
Anyone who abandons their animal is committing a punishable offense.
In the meaning 'to pause for a round', 'aussetzen' is often used without an object. It is clear that the person themselves is interrupting the activity.
Ich bin müde, ich setze bei diesem Spiel aus.
I'm tired; I'm sitting out this game.
The meaning 'to criticize' exists only in the fixed expression 'etwas an jemandem/etwas auszusetzen haben' (to have something to find fault with in someone/something). The preposition 'an' is always followed by the dative.
Meine Mutter hat immer etwas an meiner Kleidung auszusetzen.
My mother always has something to find fault with in my clothes.
When you 'aussetzen' (expose) someone or something to something, that thing is in the dative. The structure is: [accusative] [dative] aussetzen.
Der Bergsteiger setzte sich bewusst der Gefahr aus.
The mountaineer deliberately exposed himself to the danger.
The meaning 'to temporarily lift or suspend' is used almost exclusively in legal or official contexts. Typical objects are 'eine Strafe' (a penalty/sentence) or 'ein Urteil' (a judgment).
Die Vollstreckung des Urteils wurde vorläufig ausgesetzt.
The enforcement of the judgment was temporarily suspended.
to fit / suit / pass
In a game, passen means consciously choosing not to take a possible action on the current turn. Aussetzen, by contrast, means not taking part in a turn or an entire round, either voluntarily or because the rules require it.

Wegen der Strafkarte muss ich in diesem Zug aussetzen.
Because of the penalty card, I have to sit out this turn.
Ich möchte keine Karte legen und passe in diesem Zug.
I don't want to play a card, so I pass this turn.