{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","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-asymptotic-resource-bound"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-asymptotic-resource-bound-01","title":{"zh-Hant":"漸近資源界：證明 3n²+5n+2∈Θ(n²) 並排除 Θ(n)","en":"Asymptotic resource bounds: proving 3n²+5n+2∈Θ(n²) and ruling out Θ(n)"},"summary":{"zh-Hant":"設成本函數 f(n)=3n²+5n+2（n 從 0 起），比較函數 g(n)=n² 與 h(n)=n。以明示的全尾段 witness 證明 f∈O(g)（c=10、N=1）、f∈Ω(g)（c=3、N=0），故 f∈Θ(g)；說明 N=0 對 O 不可行、c=3 對 O 不可行、另一組 witness c=4、N=6 也可行；再證明 f∉O(h)，因此 f∈Ω(h) 但 f∉Θ(h)。","en":"Let the cost function be f(n)=3n²+5n+2 (n starting at 0), with comparison functions g(n)=n² and h(n)=n. With explicit full-tail witnesses prove f∈O(g) (c=10, N=1) and f∈Ω(g) (c=3, N=0), hence f∈Θ(g); show that N=0 cannot work for O, that c=3 cannot work for O, and that another witness c=4, N=6 also works; then prove f∉O(h), so f∈Ω(h) but f∉Θ(h)."},"mko_ids":["mko-asymptotic-resource-bound"]}],"title":"概念推導導讀"}