一、Prove by mathematical induction that

\[ (1+x)^n = C_0^n + C_1^n \times x + C_2^n \times x^2 + \dots + C_n^n \times x^n, n \in \mathbb{N}. \]
(20%)

(1) Is (p∨ q) ∧ (¬p ∨ r) ⟶ (q ∨ r) a tautology (i.e., “always true”)? Please explain your answer. Note that ¬? denotes “not p” and ⟶ denotes “p implies q”. (10%)

(2) Please prove that (p ∨ ¬q) ∧ (¬p ∨ q) and (¬q ∧ ¬p) ∨(p ∧ q) are logically equivalent by using only laws of propositional logic. (10%)

三、Please calculate the number of positive integers greater than 1 (> 1) and not exceeding 150 (≤ 150) that are divisible by none of 3, 5, or 7 by using The Principle of Inclusion–Exclusion. You should explain your answer. (10%)

四、Paul has two coolers. Cooler 1 contains 8 cans of cola and 3 cans of lemonade. Cooler 2 contains 5 cans of cola and 7 cans of lemonade. Paul randomly selects one can from Cooler 1 and transfers it into Cooler 2. Subsequently, Betty randomly selects two cans from Cooler 2. Given that both of Betty's selections are cans of cola, what is the probability that the can Paul initially transferred was lemonade? (10%)

五、Solve the following recurrence relation:an + 6an-1− 16an-2 = 6, given that a0 = 0 and a1 = 2.

六、Determine whether the poset (Partially Ordered Set) (S, ⊆) is alattice where ? ={∅, {1}, {2}, {4}, {1,2}, {1,4}, {3,4}, {1,2,4},{2,3,4}, {1,2,3,4}}Please draw its Hasse diagram and explain your answer. (10%)

(1) Let G = (V, E) be an undirected graph with d components, where |V| = b and |E| =c  . Please prove that c ≥ b − d . (10%)

(2) Consider the complete bipartite graphKm,n , where the vertex set is partitioned into two disjoint sets V1 and V2with |V1 | = m and |V2 | = n. Derive the necessary and sufficient conditions for ? and ? such that Km，n contains an Eulerian path but NO Eulerian circuit. (Consider all possible cases for m and n ). (10%)

第一題： 依職業安全衛生署公告之「高空工作車使用安全管理指引」，請說明高空工作車之選用應考量項目【16 分】及高空作業車操作人員應具何項資格及哪些能力？【9 分】

第二題： 施工人員經常使用合梯進行高架工作，依「職業安全衛生設施規則」，雇主對於使用之合梯，應符合哪些規定？【25 分】

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