國立臺中教育大學 98 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 ※問答題 (100%，每題 10%)
1. For proving implications p→q, we may use one of the following methods:
a. Prove ¬p by itself.
b. Assume ¬q, and prove ¬p.
c. Prove q by itself.
d. Assume p is true, and prove q.
e. Proceed by exhausting all possibilities.
(1) Which one is the trivial proof? (2%) (2) Which one is the direct proof? (2%)
(3) Which one is the indirect proof? (2%) (4) Which one is the exhaustive proof? (2%) (5) Which one is the vacuous proof? (2%)
2. Let A = {a, b} and B = {c, d, e}. Find the power set of the Cartesian product B × A.
(1) Show the Cartesian product A × B. (5%)
(2) Show the number of elements within the power set of the Cartesian product A × B. (5%)
3. Let R be the set of real numbers. Determine whether the function f(x) = x2 from R to R is onto? Justify your answer.
4. Suppose that a cake shop has 5 different kinds of cakes, and there are 10 pieces of each kind of cakes. How many different ways can 4 cakes be chosen from this cake shop?
5. Suppose that the 0-1 matrix of the relation R is ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = 1 0 1 0 1 1 1 0 0 R M
(1) Find the 0-1 matrix of the reflexive closure of R. (3%) (2) Find the 0-1 matrix of the symmetric closure of R. (3%) (3) Find the 0-1 matrix of the transitive closure of R. (4%)
6. Given a weighted connected graph G1 (as shown in Figure 1), please write down one of its minimum cost spanning trees and its total cost.
Figure 1: G1
7. Write down one of the circuit if any Eular circuit exists in G2 (as shown in Figure 2) and G3 (as shown in Figure 3). Also, please explain your reason.
Figure 2: G2 Figure 3: G3
8. Given a weighted connected graph G4 (as shown in Figure 4), please write down the shortest path between Vertex 1 and Vertex 5. Also, please write down the procedure that you find out the shortest path.
Figure 4: G4
9. Given a connected simple graph that has n vertices. (a) Please write down the minimal number of edges of this graph. (b) Please write down the number of edges of the minimal spanning tree of this graph.
10. Given a connected planar having 5 vertices whose degree is 3, 2, 2, 3, and 2 respectively. (a) Please write down the number of edges of this planar. (b) Please write down the number of regions of this planar.
