{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"paths","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":{},"path":{"id":"path-pulse-foundations-20261017","title":{"zh-Hant":"七輪離散結構：雙射、反函數、關係、圖樹與反例","en":"Seven Rounds of Discrete Structure: Bijections, Inverses, Relations, Graphs, Trees, and Counterexamples"},"summary":{"zh-Hant":"在七個既有知識物件及其前置上深化跨域題族；保持數學來源不變，以精確有限切片核對邊界。","en":"Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-injective","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-surjective","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-composition","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-linear-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-set-membership","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-rational-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-proof","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-bijective","position":2,"requirement":"optional","rationale":{"zh-Hant":"完整Fin總表把單射、滿射與雙射分開；全纖維／碰撞／漏項、交換關係資格、交換兩個輸出標籤、基數存在性與給定表分開。相同基數不審定給定f；空到空雙射真。","en":"Validate complete finite total maps before separating injectivity, surjectivity and bijectivity. Enumerate fibers, collisions and missing targets, qualify the converse relation, transpose output labels and distinguish cardinality equality from the given map. Empty-to-empty is bijective."}},{"id":"mko-relation","position":2,"requirement":"optional","rationale":{"zh-Hant":"R是明示Fin來源×目標的有序對集合：完整投影／漏源、子集合像與全纖維、逆關係再逆、單值與總表分開。關係可多值或缺源；重複對與集合列項正規化，逆關係不自動給反函數。","en":"Use ordered-pair sets with explicit finite source and target: projections and missing sources, restricted images and fibers, double converse, single-valuedness and totality. Normalize repeated pairs; a relation may be multivalued or partial and its converse is not automatically an inverse function."}},{"id":"mko-order-relation","position":2,"requirement":"optional","rationale":{"zh-Hant":"R通常全序的正原分母有理切片：有序三筆穩定排序、兩閉區間交會與端點、三個會員與有號餘量、純量正向／反向／零合併。要求實際新數值，不只換平移參數重出相同比較；不外推偏序或複數。","en":"Use the usual real total order on rational inputs with positive original denominators. Compute stable three-value sorts, inclusive interval intersections and membership slacks, and actual scaled values with positive/negative/zero direction guards. Do not extrapolate to complex numbers or arbitrary partial orders."}},{"id":"mko-counterexample","position":2,"requirement":"optional","rationale":{"zh-Hant":"明示有限Fin論域與完整布林述詞表：全失敗集及最小反例、三個候選的域資格、限制子域後見證是否仍有效、明示加入一個新述詞值。域外不查P也不當反例；空域全稱量化真，有限枚舉不證無限域。","en":"Use a declared finite domain and complete Boolean predicate table. Enumerate failures and least witnesses; guard candidates by the domain, recheck witnesses under restriction, and explicitly provide any new predicate value on extension. Outside inputs are not counterexamples; empty universal quantification is true and finite enumeration proves no infinite-domain claim."}},{"id":"mko-inverse-function","position":3,"requirement":"optional","rationale":{"zh-Hant":"完整Fin總表先雙射再建反表；候選g逐格合型後才核兩側identity；查詢先陪域資格。不把反函數當函數值倒數。R到R仿射恢復限非零斜率，兩個合法有理輸入以最簡正分母回代。","en":"Construct inverse tables only after bijectivity; type every candidate inverse cell before checking both identities independently. Guard queries by declared codomain. An inverse is not a reciprocal; affine recovery on R requires nonzero slope and exact normalized rational substitution."}},{"id":"mko-graph","position":3,"requirement":"optional","rationale":{"zh-Hant":"簡單無向Fin圖：完整鄰接矩陣／度數與握手、全頂點鄰居／非鄰居分割、有效邊toggle的度數改變、明示子集合誘導圖且保原標籤。反向或重複邊不增加邊，無自環；本graph輪不作連通或樹判定。","en":"Use complete simple undirected finite graphs: adjacency matrices and degree sums, neighbor/nonneighbor partitions, valid edge toggles and induced subsets with original labels. Reversed or repeated edges count once; exclude loops and do not infer adjacency from drawings or claim tree qualification here."}},{"id":"mko-tree","position":4,"requirement":"optional","rationale":{"zh-Hant":"非空樹資格與完整簡單無向圖資料：components／有環與空域互達、原樹與端點資格後唯一道路、度1葉刪除保原標籤、原樹上新增不存在邊的唯一環。空圖不是樹；單點無邊是樹但度0非葉，邊數不單獨判樹。","en":"Require a nonempty connected acyclic simple undirected graph. Check components/cycles, original-tree and endpoint guards before paths, degree-one leaf deletion without relabeling, and a new nonedge’s unique cycle. Empty graphs are not trees; a singleton is a tree but its degree-zero vertex is not a leaf. Edge count alone does not qualify a tree."}}]},"title":"Seven Rounds of Discrete Structure: Bijections, Inverses, Relations, Graphs, Trees, and Counterexamples"}