{"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-logical-connectives"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-logical-connectives-01","title":{"zh-Hant":"邏輯連接詞：以整除命題檢驗 ∨、→、↔ 與括號","en":"Logical connectives: testing ∨, →, ↔ and brackets on divisibility statements"},"summary":{"zh-Hant":"對整數 n，令 P(n) 為「4∣n」，Q(n) 為「2∣n」。(1) 判定「4∣12 ∨ 6∣12」的真假，說明析取是含括或；(2) 證明 ∀n∈ℤ (P(n)→Q(n))；(3) 證明 ∀n∈ℤ (Q(n)→P(n)) 為假，因而 ∀n∈ℤ (P(n)↔Q(n)) 為假；(4) 判定 P(3)→Q(3) 與「2+2=4 → 7 是奇數」的真假，說明蘊涵不是因果；(5) 比較 ¬Q(2)∧P(2) 與 ¬(Q(2)∧P(2)) 的真假。","en":"For an integer n let P(n) be '4∣n' and Q(n) be '2∣n'. (1) Decide the truth of '4∣12 ∨ 6∣12' and explain that disjunction is inclusive; (2) prove ∀n∈ℤ (P(n)→Q(n)); (3) prove that ∀n∈ℤ (Q(n)→P(n)) is false, and hence that ∀n∈ℤ (P(n)↔Q(n)) is false; (4) decide the truth of P(3)→Q(3) and of '2+2=4 → 7 is odd', explaining that implication is not causation; (5) compare the truth of ¬Q(2)∧P(2) and ¬(Q(2)∧P(2))."},"mko_ids":["mko-logical-connectives"]}],"title":"Concept derivation guides"}