set down has 11 distinct synonyms spread across 5 of its 6 senses. Each sense below keeps its own list, because a synonym belongs to a meaning and not to a word: swapping one sense's substitute into another sense's sentence is how a thesaurus makes writing worse. WordNet records no antonym for it — antonymy is a relation between word forms rather than concepts, so it is deliberately sparse.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
put down in writing; of texts, musical compositions, etc.
put or settle into a position
“The hotel was set down at the bottom of the valley”
cause to sit or seat or be in a settled position or place
“set down your bags here”
Synonyms, antonyms and the broader and narrower relations for set down come from Princeton University’s WordNet. Full sources and licences.