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 toa) 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 thatcontains 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 isa) 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 . Whichof 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) 11)(2-=n n fII. 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. ()h 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 proposition “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 1i 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)} andf ={(1,3),(2,2),(3,4),(4,2)}.Then fg = . 7.The sum of 2 + 4 + 8 + 16 + 32 + ... + 210 is .8. The expression of gcd(45, 12) as a linear combination of 12 and 45 is . 9. There are permutations of the seven letters A,B ,C ,D ,E ,F have A immediately to the left of E .10.If G is a planar connected graph with 18 vertices, each of degree 3, then G has _ __ regions.IV. Answer the Questions (32%)：1. Determine whether the following argument is valid: p → r q → r q ∨ ⌝r ________∴ ⌝p2. Suppose you wish to prove a theorem of the form “if p then q ”. (a) If you give a direct proof, what do you assume and what do you prove? (b) If you give an indirect proof, what do you assume and what do you prove? (c) If you give a proof by contradiction, what do you assume and what do you prove?3. Prove that A B A B ⋂=⋃ by giving a proof using logical equivalence.4.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).e the definition of big-oh to prove that5264473n nn+--is O(n3).6.Solve the linear congruence 5x≡ 3 (mod 11).e the Principle of Mathematical Induction to prove that131 1392732nn+-++++...+=for alln≥ 0.8.Draw the directed graph for the relation defined by the matrix1111 0111 0011 0001⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦.VI.(6%) Devise an algorithm which will find the minimum of n integers. What is the worst case time complexity of this algorithm?VII.(5%) Give the definition of a transitive relation, and Prove or disprove that the union of two transitive relations is transitive.VIII. (6%) The pseudo-code of Prim ’s algorithm is given as following:Procedure Prim (G : connected weighted undirected graph with n vertices) T := a minimum-weight edge for i := 1 to n 2 begine := an edge of minimum weight incident to a vertex in T and not forming a simple circuit in T if added to T T := T with e addedPrint eend {T is a minimum spanning tree of G }(a) Find a minimum spanning tree using Prim ’s algorithm given above. For every iterative in for-loop, list theresult for “Print e ” statement.(b) Compute the total weight of the spanning tree.。