{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-surjective"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-surjective-01","title":{"zh-Hant":"滿射與陪域覆蓋：值域、原像與陪域變更","en":"Surjectivity and codomain coverage: range, preimages and changing the codomain"},"summary":{"zh-Hant":"設 A={1,2,3,4,5}、B={1,2,3}，函數表 f：(1,2),(2,1),(3,3),(4,2),(5,1)。確認 f 是總函數 A→B，求值域 f[A] 與各原像，判定 f 是否滿射並找出 t=2 的首個原像；再把陪域換成 B2={1,2,3,4} 與 B3={1,2} 判定合法性與滿射性；最後處理 ∅→∅ 與 ∅→{7}。","en":"Let A={1,2,3,4,5}, B={1,2,3} and the function table f: (1,2),(2,1),(3,3),(4,2),(5,1). Confirm f is a total function A→B, find the range f[A] and the preimages, decide whether f is surjective and find the first preimage of t=2; then replace the codomain by B2={1,2,3,4} and by B3={1,2}, deciding legality and surjectivity; finally treat ∅→∅ and ∅→{7}."},"mko_ids":["mko-surjective"]}],"title":"Concept derivation guides"}