离散数学期末考试_2013年春季_试卷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)（ ） 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 havea) 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 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. （ ） 4.The two's complement of -13 isA. 1 0011.B. 0 1101 a)C. 1 0010D. 0 1100（ ） 5. How many bit strings of length 10 have exactly six 0’s?a) 210 b) C(10,6). c) 26 d) 36（ ） 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. 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) bad) a （ ） 8.Which of the following set is uncountable?a) The set of real numbers between 172 and 173. b) The set of integersc) 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. .A ⋃ (B ⋂ C ) ⊇ (A ⋃ B ) ⋂ C .（ ） 9. The set {∅,{a },{∅,a }} is the power set of some set.（ ） 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 1i 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.Write the negation of the stat ement “No tests are easy ” in good English:. 6. The expression of GCD(45,12) as a linear combination of 12 and 45is . 7. There are 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) = . 9.The negation of the statement ∃x ∀y (xy = 0) is.10. Let }|),{(},|),{(2221b a b a R b a b a R ≠ℜ∈=≤ℜ∈=Then 21R R ⋂ is.IV. Answer the Questions (35%,7 questions, 5 points each)：1.Write the truth table for the proposition：⌝(r→⌝q) ∨ (p∧⌝r)2.Suppose f:R→R where f(x) =⎣x/2⎦.(a) If S={x| 1 ≤x≤ 6}, find f(S).(b) If T={3,4,5}, find f-1(T).3.Find the matrix that represents the relation of R on {1,2,3,4}where aRb means |a-b|≤1. Useelements in the order given to determine rows and columns of the matrix.4.p,q,⌝, and the connective ∨ that has the given truth table.5.Encrypt the message “HELP” by translating the letters into numbers, applying the encr yption functionf (p) = (3p+ 7) mod 26, and then translating the numbers back into letters.6.Solve the linear congruence 5x≡ 3 (mod 11).7. In the questions below 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)}. Find f g .V. (6%) Prove that (q ∧ (p → ⌝q )) → ⌝p is a tautology using propositional equivalenceand the laws of logic.VI. (7%) Let A be the set of all points in the plane with the origin removed. That is,A = {(x, y ) | x, y ∈ R } − {(0, 0)}.Define a relation on A by the rule: (a, b )R (c, d ) (a, b ) and (c, d ) lie on the same line through the origin. (a) Prove that R is an equivalence relation.(b) Describe the equivalence classes arising from the equivalence relation R in part (a). (c) If A is replaced by the entire plane, is R an equivalence relation?VII.(7%) Determine whether this argument is valid:Lynn works part time or full time.If Lynn does not play on the team, then she does not work part time. If Lynn plays on the team, she is busy.Lynn does not work full time.Therefore, Lynn is busy.。