99年私立中原大學資工所離散數學

Complete the blanks in the following problems:

1.

The number of integer solutions to the following equation x₁ + x₂ + x₃ = 17 with x₁ ≥ 1, x₂ ≥ −2, x₃ ≥ 3 is _______.

Complete the blanks in the following problems:

2.

The representation of the arithmetic expression involving the operators ？ (multiplication), / (division), − (subtraction), and ↑ (exponentiation), the value of the postfix expression 8 3 2 ？ − 4 ↑ 9 3 / + is _______

Complete the blanks in the following problems:

3.

Letfor k = 0, 1, 2, …. The generating function for the sequence a₀, a₁, a₂, … is ______.

Complete the blanks in the following problems:

4.

Let P(x) and Q(x) be the statements "x is a lion" and "x is fierce," respectively. If the domain consists of all creatures, using quantifiers, logical connectives and P(x) and Q(x), the statement "All lions are fierce." can be expressed as _______.

Complete the blanks in the following problems:

5.

log(4n³ + n!) + sin(n²) + 10n = O(_______).

6.

Let P(x), Q(x), R(x) be open statements that are defined for a given universe.
Show that

is valid.

7.

Find the solution to the recurrence relation
a_n − 2a_n−1 − 2a_n−2 + 4a_n−3 + a_n−4 − 2a_n−5 = 32.

8.

Let a₁, a₂, …, a_n be a sequence of positive integers. Show that there exist two positive integers k and m, 1 ≤ k < m ≤ n such that the sum a_k+1 + a_k+2 + ... + a_m is divisible by n.

Let A = {2, 3, 6, 12, 24, 36} and R be a relation on A, R = {(x, y) | x ∈ A, y ∈ A, x divides y}.

9.

Find R.

Let A = {2, 3, 6, 12, 24, 36} and R be a relation on A, R = {(x, y) | x ∈ A, y ∈ A, x divides y}.

10.

Show that R is a partial ordering relation.

Let A = {2, 3, 6, 12, 24, 36} and R be a relation on A, R = {(x, y) | x ∈ A, y ∈ A, x divides y}.

11.

Find the minimal element, maximal element, greatest lower bound (glb), least upper bound (lub) of A.