{"id":"function","domain":"foundations-logic","type":"Concept","title":"Function","slug":"function","url":"/mathematics/foundations-logic/function/","summary":"A rule assigning exactly one output to each input from a given set. Formally a subset of A×B satisfying uniqueness. Fundamental across all mathematics.","formal_definition":"f:A→B is a subset of A×B such that for every a∈A there is exactly one b∈B with (a,b)∈f.","key_properties":["Injective (one-to-one)","Surjective (onto)","Bijective (both, has inverse)"],"added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Enderton, H.B. (1977). Elements of Set Theory. Academic Press."},{"tier":1,"citation":"Kline, M. (1972). Mathematical Thought from Ancient to Modern Times. Oxford University Press."},{"tier":2,"citation":"Mac Lane, S. (1998). Categories for the Working Mathematician. 2nd ed. Springer."}],"relationships":[{"type":"built_from","target":"set","label":"Set"},{"type":"used_in","target":"limit","label":"Limit"},{"type":"used_in","target":"vector-space","label":"Vector Space"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.4"}