
festsetzen
to set
To determine, establish, or define something in a binding way. This often concerns prices, dates, rules, or penalties.
Das Gericht wird die Höhe der Strafe morgen festsetzen.
The court will set the amount of the fine tomorrow.
to set / settle / detain
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to set
To determine, establish, or define something in a binding way. This often concerns prices, dates, rules, or penalties.
Das Gericht wird die Höhe der Strafe morgen festsetzen.
The court will set the amount of the fine tomorrow.

to settle
To stick to a place or accumulate there so that it is difficult to remove.
Der hartnäckige Schmutz hat sich in den Fugen der Fliesen festgesetzt.
The stubborn dirt has settled in the joints between the tiles.

to detain
To arrest a person and hold them in a place; to take someone into custody.
Die Polizei konnte die Verdächtigen nach einer kurzen Verfolgung festsetzen.
The police were able to detain the suspects after a short chase.

to take hold
In a figurative sense: when an idea, thought, or feeling becomes anchored in one's consciousness and no longer disappears.
Der Gedanke, dass er einen Fehler gemacht hatte, hatte sich in seinem Kopf festgesetzt.
The thought that he had made a mistake had taken hold in his mind.
'Festsetzen' (to set / settle / detain) is a separable verb. The prefix 'fest-' comes at the end in a main clause; in subordinate clauses it remains attached to the verb. In the infinitive with 'zu', 'zu' comes between 'fest' and 'setzen'.
Die Firma setzt die neuen Preise fest.
The company sets the new prices.
The verb consists of 'fest' (in the sense of 'binding' or 'fixed') and 'setzen' (to place). So the basic idea is to put something in a fixed, unchangeable place, whether it is a rule, a person, or a thought.
Der König setzte eine hohe Belohnung für die Ergreifung des Diebes fest.
The king set a high reward for the capture of the thief.
The meaning changes depending on whether the verb is used actively or reflexively ('sich festsetzen'). Active means that someone determines something or holds someone. Reflexive means that something sticks or becomes anchored by itself.
Die Regierung setzt die neuen Steuern fest.
The government sets the new taxes.
In the meaning 'bestimmen' (to determine), 'festsetzen' (to set) is typically used with abstract nouns. Common combinations are: 'einen Termin festsetzen' (to set a date), 'einen Preis festsetzen' (to set a price), 'eine Frist festsetzen' (to set a deadline), or 'eine Strafe festsetzen' (to set a penalty).
Die Kommission muss nun die genauen Bedingungen festsetzen.
The commission now has to set the exact conditions.
When 'festsetzen' (to detain) has a person as its object, the context is almost always police, military, or law. It then means to take a person into custody.
Die Küstenwache konnte das verdächtige Schiff auf offener See festsetzen.
The coast guard was able to detain the suspicious ship on the open sea.
to determine
Bestimmen and festsetzen can both refer to a binding decision. Bestimmen is general and often emphasizes the person or authority making the decision. Festsetzen refers to a formal, precise determination and is especially used with prices, dates, rules, or penalties.

Der Veranstalter setzt den Eintrittspreis auf zehn Euro fest.
The organizer sets the admission price at ten euros.
Der Veranstalter bestimmt, wann der Verkauf beginnt.
The organizer decides when sales will begin.