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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
> 申論題
1. (15 points) Let S be a set of n elements. Let A, B be two different subsets of S chosen uniformly al Tandom. What is the probability that A is a subset of B? Show your derivation.
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2. (10 points) Solve the following recurrence (show your derivation):
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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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