{"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":{"from":"/en/paths/"},"path":{"id":"path-pulse-foundations-20261004","title":{"zh-Hant":"每日七輪第一期：量詞、離散結構與線性／条件模型","en":"Daily Seven-Round Cycle I: Quantifiers, Discrete Structures, Linear Systems, and Conditional Models"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-proposition","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-proof","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-membership","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-coordinate-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-linear-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-probability","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quantifier","position":2,"requirement":"optional","rationale":{"zh-Hant":"全稱與存在量詞必須綁定論域；古典真值語義下，空論域的全稱為真、存在為假。","en":"Universal and existential quantifiers must bind a domain. Under classical truth semantics, a universal statement over the empty domain is true and an existential statement is false."}},{"id":"mko-counterexample","position":3,"requirement":"optional","rationale":{"zh-Hant":"反例必須屬於被聲稱的論域且使全稱述詞失敗；域外值和找不到反例都不能自動證明原命題。","en":"A counterexample must belong to the claimed domain and falsify the universal predicate. An outside value or failure to find a counterexample does not by itself prove the original statement."}},{"id":"mko-relation","position":4,"requirement":"optional","rationale":{"zh-Hant":"二元關係是指定 A×B 中有序對的集合，不要求每個輸入唯一或必有輸出。","en":"A binary relation is a set of ordered pairs in a specified A×B. An input need not have a unique output or any output."}},{"id":"mko-graph","position":5,"requirement":"optional","rationale":{"zh-Hant":"本輪限定有限簡單無向圖，邊是兩個相異頂點的無序集合，度數只計相鄰頂點。","en":"This entry covers finite simple undirected graphs. An edge is an unordered set of two distinct vertices, and degree counts distinct neighbors."}},{"id":"mko-vector","position":6,"requirement":"optional","rationale":{"zh-Hant":"本輪把平面向量限制為固定基底下的 R² 有序對，逐分量定義加法與純量乘法。","en":"This entry represents plane vectors as ordered pairs in R² in a fixed basis, defining addition and scalar multiplication componentwise."}},{"id":"mko-system-linear-equations","position":7,"requirement":"optional","rationale":{"zh-Hant":"同時滿足兩個實數線性限制才是解；非零行列式給唯一解，奇異與零列須另判一致性。","en":"A solution must satisfy two real linear constraints simultaneously. A nonzero determinant gives a unique solution; singular systems and zero rows require separate consistency checks."}},{"id":"mko-conditional-probability","position":8,"requirement":"optional","rationale":{"zh-Hant":"條件事件必須有正機率；有限模型中 P(A|B)=P(A∩B)/P(B)，方向不能任意交換。","en":"The conditioning event must have positive probability. In a finite model, P(A|B)=P(A∩B)/P(B), and the direction cannot be freely exchanged."}}]},"title":"Daily Seven-Round Cycle I: Quantifiers, Discrete Structures, Linear Systems, and Conditional Models"}