电子科技大学离散数学考题2013年13年A-英才

电子科技大学英才学院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)

（C ） 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) （C ） 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.

（B ） 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 Engli sh 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. (A ） 4.

The two's complement of -13 is

A. 1 0011.

B. 0 1101 a)

C. 1 0010

D. 0 1100

?（ B ） 5. How many bit strings of length 10 have exactly six 0’s?

a) 210 b) C(10,6). c) 26 d) 36

（ B ） 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

（ C ） 7. S is a collection of strings of symbols. It is recursively defined by 1) a and b belong to S;

2) if string X belongs to S, so does Xb. Which of the following does NOT belong to S? a) abbb b) bbb c) ba

d) a （ A ） 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. （ B ） 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

?（C ） 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)

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

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

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

（F ）

7.

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

?（T ） 8. .A ⋃ (B ⋂ C ) ⊇ (A ⋃ B ) ⋂ C .

（F ） 9. The set {∅,{a },{∅,a }} is the power set of some set. ?（F ） 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 2^7 .

2.

If 11

{|}i A x x R x i i =∈∧-≤≤ then 1

i i A +∞

= is [-1,1] .

？

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 =

(1,2)(2,3)(3,3)(4,2) . 5.

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

Not every text is easy . . ？6. The expression of GCD(45,12) as a linear combination of 12 and 45

is .

7. There are 720 permutations of 7letters A ,B ,C ,D ,E ,F ,G have A immediately to the left of E . 8. If f (n ) = f (n - 1) / f (n - 2), f (0) = 2, f (1) = 5, Then f (2) = 2.5 . ？9.

The negation of the statement ∃x ∀y (xy = 0) is

.

10. Let }|),{(},|),{(22

21b a b a R b a b a R ≠ℜ∈=≤ℜ∈=

Then 21R R ⋂ is

(a,b),a

.