renunciation / waiver
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renunciation
The voluntary or forced decision to give something up, such as a claim, a right, or a pleasure.
Der Verzicht auf sein Erbe war eine schwere Entscheidung für ihn.
The renunciation of his inheritance was a difficult decision for him.
'Der Verzicht' (renunciation/waiver) is the noun that belongs to the verb 'verzichten' (to do without / to give up). When you do without something, the result is 'der Verzicht' (renunciation).
Ich verzichte heute auf den Nachtisch.
I'm skipping dessert today.
The noun is almost always used with the preposition 'auf' and the accusative: 'der Verzicht auf etwas' (the renunciation of something / doing without something).
Der Verzicht auf Plastiktüten ist ein wichtiger Schritt für die Umwelt.
Doing without plastic bags is an important step for the environment.
A common construction describes how hard or easy the action is for a person: 'Der Verzicht fällt jemandem (dative) schwer/leicht' (giving something up is hard/easy for someone).
Nach so vielen Jahren fiel ihm der Verzicht auf das Rauchen sehr schwer.
After so many years, giving up smoking was very hard for him.
victim / sacrifice
“Verzicht” refers more neutrally to giving something up or not using it. “Opfer” emphasizes more strongly that what is given up is personally important or valuable and that the decision serves a higher goal or other people.
Ihr Verzicht auf die Beförderung war eine freiwillige Entscheidung.
Giving up the promotion was her own voluntary decision.
Für ihre Familie brachte sie das Opfer, auf die Beförderung zu verzichten.
For her family, she made the sacrifice of giving up the promotion.
