双语离散数学期末考试_2012年春季_试卷A

电子科技大学2011 -2012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： 2012 年 6 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%)

1. (⌝p ∧q)→(p ∨q) is logically equivalent to

a) T b) p ∨q c) F d) ⌝ p ∧q （ ） 2. If P(A) is the power set of A, and A = , what is |P(P(P(A)))|?

a) 4 b) 24 c) 28 d) 216

（ ） 3. Which of these statements is NOT a proposition?

a) Tomorrow will be Friday. b) 2+3=4.

c) There is a dog. d) Go and play with me.

（ ）

4. The notation K n denotes the complete graph on n vertices. K n is the simple graph that

contains exactly one edge between each pair of distinct vertices. How many edges comprise a K 20?

a) 190 b) 40 c) 95 d) 380

（ ）

5. Suppose | A | = 5 and | B | = 9. The number of 1-1 functions f : A → B is

a) 45 b) P (9,5). c) 59 d) 95

（ ）

6. Let R be a relation on the positive integers where xRy if x divides y . Which

of the following lists of properties best describes the relation R ?

a) reflexive, symmetric, transitive b) reflexive, antisymmetric, transitive c) reflexive, symmetric, antisymmetric d) symmetric, transitive （ ）

7. Which of the following are partitions of }8,7,6,5,4,3,2,1{=U ?

a) }8,7,6,5,4,3{},3,2,1{},1{ b) }8,7,6,5,4,3{},3,2{},1{

c) }8,6,5{},3,2{},7,4,1{ d) }8,7,6,5,4{},3,2{},2,1{

（ ） 8. The function f(x)=3x 2log(x 3+21) is big-O of which of the following functions? a) x 3 b) x 2(logx)3 c) x 2logx d) xlogx （ ） 9.

In the graph that follows, give an explanation for why there is no path from a back to a that passes through each edge exactly once.

a) There are vertices of odd degree, namely {B,D}. b) There are vertices of even degree, namely {A,C}. c) There are vertices of even degree, namely {B,D}. d) There are vertices of odd degree, namely {A,C}.

（ ） 10. Which of the followings is a function from Z to R ?

a) )1()(-±=n n f . ` b) 1)(2

+=x x f . c) x x f =

)( d) 1

1

)(2-=

n n f

II. True or False (10%)

（ ） 1. If 3 < 2, then 7 = 6. （ ） 2. p ∧ (q ∨ r)≡ (p ∧ q) ∨ r

（ ） 3. If A , B , and C are sets, then (A -C )-(B -C )=A -B . （ ） 4. Suppose A = {a ,b ,c }, then {{a }} ⊆ P (A ).

（ ） 5. ()100h x x =+is defined as a function with domain R and codomain R.

（ ） 6. Suppose g : A → B and f : B → C , where f g is 1-1 and f is 1-1. g must be 1-1? （ ） 7. If p and q are primes (> 2), then p + q is composite .

（ ） 8.

If the relation R is defined on the set Z where aRb means that ab > 0, then R is an equivalence relation on Z .

（ ） 9. Every Hamilton circuit for W n has length n .

（ ） 10. There exists a simple graph with 8 vertices, whose degrees are 0, 1, 2, 3, 4, 5, 6, 7.

III. Fill in the Blanks (20%)

1. Let p and q be the propositions “I am a criminal” and “I rob banks”. Express in simple English the propositi on “if p then q”: .

2. P (x ,y ) means “x + 2y = xy ”, where x and y are integers. The truth value of ∃x ∀yP (x ,y ) is .

3. T he negation of the statement “No tests are easy.” is .

4. If 11

{|}i A x x R x i i =∈∧-≤≤ then 1

i i A +∞

=is .

5. Suppose A = {x , y }. Then ()P A is .

6. Suppose g : A →A and f :A →A where A ={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 = . 7.

The sum of 2 + 4 + 8 + 16 + 32 + ... + 210 is .