(二)說明屋頂上設置太陽能板時的注意事項。【20 分】

 (一)說明自然通風的設計原理與方式。【15 分】

(二)說明機械通風的設計原理與方式。【10 分】

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%)

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