110年 - 110 國立中山大學_碩士班招生考試_電機系(丙組)：離散數學#104260

試卷資訊

選擇題 (2)

複選題
Problem 1. (5%) TRUE or FALSE: Decide whether or not the following statements are True(O) or False(X). You do not have to justify the answer. Each correct answer is 1 point, and each incorrect one is -3 points (until you get 0 points in this test). If you choose not to answer, you get 0 points for each. (A) If i and j belong to a ring, then (i + j)²= i² +2ij +j² (B) Every cyclic group is an abelian group but not every abelian group is a cyclic group (C) A normal subgroup is a monoid with the same number of elements of the original group (D) For any finite-dimensional commutative local algebra L over a field F, L is self-injective (E) A clique of an undirected graph G is a complete subgraph of G

Problem 2. (5%) 2003 is a prime number. Find out how many square numbers are the factors of 20032003. You will be awarding -2 points for the incorrect answer(until you get O points in all the problems of this test). Please justify your answer. Otherwise, you get O points. (A) 0 (B) 1 (C) 44 (D) 1002 (E) 2003

申論題 (6)

Problem 3. (5%) Let x = (1,3,5)(1,2) and y = (1,5,7,9) Group G. Please calculate. Please justify your answer. Otherwise, you get 0 points.

Problem 4. (15%) There are two apples and four oranges. You would like to buy three of them, but at least one apple should be selected. Please show that how many choices you have. Please justify your answer. Otherwise, you get 0points.

Problem 5 (15%) Assume that φ: R→ R is a function that satisfies

where a is a real number. Findφ(0). Please justify your answer. Otherwise, you get 0 points.

Problem 6 (15%) Given an equation xy + x + y = 2004, where x (＞0) and y (＞0) are both integers, find the solution(s). Please justify your answer. 0therwise, you get 0 points.

Problem 7 (15%) Find F(n) that satisfies the following recurrence
F(n) = 5F(n - 1) - 6F(n -2).
Please justify your answer. Otherwise, you get 0 points.

Problem 8 (25%) Apply the Depth-First Search algorithm to find out the traversal order, where the vertex "a" is the start, and the vertices with lower adjacent weights should be visited first. Please justify your answer. Otherwise, you get 0 points. Hint: Use a stack.