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國立臺中教育大學 99 學年度大學日間部轉學招生考試
離散數學試題
適用學系:資訊科學學系二、三年級
1. Suppose that the domain of the propositional function P(x) consists of the integers 5, 6, 7, and 8. Express each of the following statements without using quantifiers, instead using only negations (¬), disjunctions (∨), and conjunctions (∧).
(1) ∃x P(x) (5%) (2) ¬ ∀x P(x) (5%)
2. How many ways are there to distribute 8 indistinguishable (identical) balls into 5 distinguishable bins? (10%)
3. Find the solution to an = 7an–2 + 6an–3 with a0 = 9, a1 = 10, and a2 = 32. (10%)
4. Let R be the relation on the set {0, 1, 2, 3} containing the ordered pairs (0, 1), (1, 1), (1, 2), (2, 0), (2, 2), and (3, 0).
(1) Find the reflexive closure of R. (5%)
(2) Find the symmetric closure of R. (5%)
5. What is the minimum number of students required in a class to guarantee that at least 15 students will receive the same grade, if there are 5 possible grades, A, B, C, D, and F? (10%)
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6.For the prefix code given in Figure 1, please decode the sequence 1001111101. (15%)
Figure 1
7. Please find a Hamilton cycle if one exists, for the graph or multigraphs in Figure 2. If the graph has no Hamilton cycle, determine whether it has a Hamilton path. (15%)
Figure 2
8. Please find the minimal spanning tree by applying Kruskal’s Algorithm to the graph shown in Figure 3. Also, please write down the weight. (20%)
