电子科技大学离散期末试题13年

电子科技大学2012 -2013学年第 2学期期 末 考试 A 卷

课程名称： 离散数学 考试形式： 闭卷 考试日期： 2013 年 月 日 考试时长：120分钟 课程成绩构成：平时 10 %， 期中 20 %， 实验 0 %， 期末 70 % 本试卷试题由____ _部分构成，共_____页。

I.

Multiple Choice (15%, 10 questions, 1.5 points each)

（ ） 1.

Which of these propositions is not logically equivalent to the other three? a) (p → q) ∧ (r → q) b) (p ∨ r) → q c) (p ∧ r) → q d) ¬q → (¬p ∧¬r) （ ） 2. Suppose A = {x ,y } and B = {x ,{x }}, then we don ’t have

a) x ∈ B b)∅ ∈ P (B ). c) {x } ⊆ A - B . d)| P (A ) | = 4.

（ ） 3.

Suppose the variable x represents students, F (x ) means “x

is a freshman”, and M (x ) means “x is a math major”. Match the statement “⌝∃x (M (x ) ∧ ⌝F (x ))” with one of the English statements below:

A. Some freshmen are math majors.

B. Every math major is a freshman.

C. No math major is a freshman.

D. Some freshmen are not math majors. （ ） 4.

The two's complement of -13 is

A. 1 0011.

B. 0 1101 a)

C. 1 0010

D. 0 1100

（ ） 5.

The chromatic number of a graph is the least number of colors needed for a coloring of this graph. The chromatic number of the graph G is

a) 2 b)3

c)4 d) 5

（ ） 6. The function f(x)=x 2log(x 3

+100) is big-O of which of the following functions? a) x 2 b)x 2logx c) x(logx)3 d) xlogx

（ ） 7.

Which of the following complete graphs is planar?

a) K 5 b) K 3,3 c) K 6 d) K 4

（ ） 8.

Which of the following set is uncountable ?

a) The set of real numbers between 172 and 173. b) The set of integers

c) The set of integers not divisible by 3. d) The union of two countable sets. （ ） 9.

How many numbers must be selected from the set {2,4,6,8,10,12,14,16,18,20} in order to guarantee that at least one pair adds up to 22?

a) 5 b) 6 c) 7 d) 8

（ ） 10. Which of the following is false?

a) {x}⊆{x} b) {x}∈{x, {x}} c) {x}⊆P({x}), where P({x}) is the power set of {x} d) {x}⊆{x, {x}}

II. True or False (10%, 10 questions, 1 point each)

（ ） 1. The proposition ((p → q ) ∧ ⌝p ) → ⌝q is a tautology. （ ） 2. If pigs can fly, then it will be raining tomorrow. （ ） 3. Suppose A = {a ,b ,c }, then {{a }} ⊆ P (A ).

（ ） 4. “My daughter visited Europe last week” implies the conclusion “Someone visited Europe last week”.

（ ） 5. For all integers a ,b ,c ,d , if a | b and c | d , then (a + c )|(b + d ). （ ） 6. For all real numbers x and y , ⎣x - y ⎦ = ⎣x ⎦ - ⎣y ⎦.

（ ）

7. ()h x =is defined as a function with domain R and codomain R.

（ ） 8. There exists a simple graph with 6 vertices, whose degrees are 2,2,2,3,4,4.

（ ） 9.

If G is a simple graph with n ≥3 vertices such that the degree of every vertex in G is at least n/2, then G has a Hamilton circuit.

（ ） 10. Let P (m ,n ) be the statement “m|n ,” where the u.d . of m and n is the set of positive integer.

Then ),(n m mP n ∀∃holds.

III. Fill in the Blanks (20%, 10 questions, 2 points each)

1. Suppose A = {x | x ∈ Z and x 2 < 10}. Then ()P A is .

2.

If 11

{|}i A x x R x i i =∈∧-≤≤ then 1

i i A +∞

= is .

3.

Give a relation on {a ,b ,c } that is reflexive and transitive, but not antisymmetric.

.

4.

Suppose g : A → B and f : B → C where A = B = C = {1,2,3,4}, g = {(1,4), (2,1), (3,1), (4,2)} and f = {(1,3),(2,2),(3,4),(4,2)}. Then f g =

. 5.

W rite the negation of the statement “No tests are easy ” in good English:

. 6. The expression of GCD(45,12) as a linear combination of 12 and 45

is . 7.

There are permutations of 7letters A ,B ,C ,D ,E ,F ,G have A immediately to the left of E .