國立臺中教育大學 96 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 問答題 (100%，每題 10%)
1. Show that if five integers are selected from the first eight positive integers, there must be a pair of these integers with a sum equal to 9.
2. A palindrome is a string whose reversal is identical to the string. For example, 101 and 1001 are palindromes.
(a) How many binary strings of length 9 are palindromes? (b) How many binary strings of length 10 are palindromes?
3. Solve the equations (a)
∑
( )
(b)= n k n k 0
( )
∑
=−
n k n k k 0)
1
(
4. Assume there are only 365 birthdays (i.e., no February 29).
(a) In a set of k people chosen at random, what is the probability that 2 or more of them have the same birthday? (7%)
(b) For 2 randomly chosen people, what is the probability that they have the same birthday? (3%)
5. Let ⊕ and be two operations defined on Z14 = {0, 1, 2, 3, …, 13} such that a ⊕ b = (a + b mod 14) and a b = (a ⋅ b mod 14). Is the algebraic system (Z14, ⊕,
) a Ring? Justify your answer.
6. Let N denote the set of all natural numbers. Given the relation R = {(a, b) | a, b ∈
N and a divides b}, please prove that R is a partial order relation.
7. Let A be a set with six distinct elements. (a) How many different binary relation on
A are there? (b) How many of them are total ordering relation?
8. Find a minimum cost spanning tree of the graph shown in Figure 1. Also, write down its cost.
1 2 3 7 6 5 4 10 28 16 12 14 24 18 22 25 Figure 1
9. Please write down the Adjacency Matrix and the Linked Adjacency List of the graph shown in Figure 2.
1
2
Figure 2
10. C = {00000, 01111, 10101, 11010} is a group code of B5 and it is d-error correcting code. Please write down the minimum distance of this code and the value of d.
2
3 4
