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109年 - 109 國立臺灣大學_碩士班招生考試_電機工程研究所丙組:離散數學(B)#105860
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2. (10 points) Solve the following recurrence (show your derivation):
相關申論題
3. (15 points) Find a positive integer p such that or show that such an integer does not exist. Prove the correctness of your answer. (a)
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(b) In propositional logic, is a functionally complete set.
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(c) If A and B are two countably infinite sets, then |A| = |B|.
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(d) If S is an infinite set, then 2S must be uncountable.
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(e) If a relation R is transitive, then R2 must also be transitive.
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(f) The set is a partial ordering on the set of all positive functions f: N→ R+.
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(g) If R1and R2 are two different relations defined on set A, then the (directed) graphs repre- senting R1 and R2 must not be isomorhpic.
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5.(10 points) Let G = (V, E) be a simple planar undirected graph with every vertex ha 5. Is it true that G must have at least 12 vertices? Prove your answer.
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6. (15 points) If a graph G has chromatic number k, but every graph G' resulting from removing one cdge from G has chromatic number at most k–1. Is it always true that every vertex in G has degree at least k–1? Prove your answer. Recall that the chromatic number of a graph is the minimum number of colors required to color all vertices such that adjacent vertices have different colors.
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一、試以新式標點斷句抄寫之。20% 古今之譏國志與通鑒者殆於肆口而罵詈則不知起古人於九原肯吾心服否邪陳氏生於西 晉司馬生於北宋茍黜曹魏之禪讓將置君父於何地而習與朱子則固江東南渡之人也惟恐 中原之爭天統也此說前人已言諸賢易地則皆然未必識遜今之學究也是則不知古人之世 不可妄論古人文辭也知其世矣不知古人之身處亦不可以遽論其文也身之所處固有榮辱 隱顯屈伸憂樂之不齊而言之有所為而言者雖有子不知夫子之所謂況生千古以後乎聖門 之論恕也己所不欲勿施於人其道大矣今則第為文人論古必先設身以是為文德之恕而已 爾（章學誠《文史通義》 ）
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