國立臺中教育大學九十七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
※ 共 10 題，每題 10 分。
1. Construct a truth table for each of the following compound propositions. (a) p ∧ ¬q
(b) (p ∧ q )→ r
2. Give a proof by contradiction of the theorem “If 3n+2 is odd, then n is odd.”
3. Use mathematical induction to prove that 1 + 3 + 5+．．．+(2n-1)= n2 whenever n is a positive integer.
4. How many solutions does the equation x1+x2+x3 =9 have, where , , and
are nonnegative integers?
1
x x2 x3
5. (a) What is the Pigeonhole Principle?
(b) Show that if six integers are selected from the first ten positive integers, there must be a pair of these integers with a sum equal to 11.
6. Let n be a nonnegative integer. Show that .
∑
2
n n k kk
n
3
0=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=7. How many different strings can be made by reordering the letters of the word “tomorrow”?
8. Show the adjacent matrix of K4, which denotes the complete graph on 4 vertices.
Next, find the number of paths of length 3 between two different vertices in K4 by
using its adjacent matrix.
9. Suppose that a connected planar graph has e edges and v vertices with v ≥ 3 and no circuits of length 3. Prove that e ≤ 2v – 4.
10. How many vertices and how many leaves does a complete 3-ary tree of height 5 have?
