{"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-injective"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-injective-01","title":{"zh-Hant":"單射與不同輸入的辨識：函數表的資格、碰撞見證與纖維","en":"Injectivity and distinct inputs: table qualification, collision witnesses and fibers"},"summary":{"zh-Hant":"設 A={1,2,3,4}、B={1,2,3,4,5,6}，函數表 f：(1,4),(2,6),(3,1),(4,5)；g：(1,4),(2,6),(2,6),(3,4),(4,6)；h：(1,4),(2,6),(2,5),(3,1),(4,5)。確認各表是否為總函數 A→B；判定 f、g 是否單射並列出纖維；找出 g 按字典序的首碰撞；判定 g 限制到 C={1,2} 是否單射；並說明空定義域的情形。","en":"Let A={1,2,3,4}, B={1,2,3,4,5,6}, with function tables f: (1,4),(2,6),(3,1),(4,5); g: (1,4),(2,6),(2,6),(3,4),(4,6); h: (1,4),(2,6),(2,5),(3,1),(4,5). Decide which tables are total functions A→B; decide whether f and g are injective and list their fibers; find g's first collision in lexicographic order; decide whether g restricted to C={1,2} is injective; and treat the empty domain."},"mko_ids":["mko-injective"]}],"title":"Concept derivation guides"}