信计11级 离散数学A试题 B卷
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negative”? ( )
A.6is even or –5 is not negative. B.6is odd and –5 is not negative.
C.6is odd or –5 is not negative. D.6is even and –5 is negative.
2. When the propositionp(qr) is true? ( )
A.Ris antisymmetric B.Ris reflexive
C.Ris symmetric D.Ris asymmetric
5. Let Q(x) be the statement “x+1<3”. If the universe of discourse consists of the integers, in the following, which is false? ( )
A . 0 B. 1 C. 2 D. 3
9.In the Boolean algebra D30, the complement (补元) of 3 is ( )
A. 10 B.6 C. 15 D. 30
山东建筑大学试卷共5页第3页
10. Let Z4be a group with operationas the Table 1, which is the inverse of [3]? ( )
9. Is the poset A={2, 3, 6, 18} under the relation of divisibility a lattice? _______.
10.Supposef:NNhas the rulef(ቤተ መጻሕፍቲ ባይዱ)=3n+2. Determine whetherfis one-to-one? ___________.
3.LetA={1，2，…，30}, define a relationRonAbyaRbif and only if |a-b|5(a,bA), determine whetherRis symmetric.
4. LetA={a,b,c},R={ (b,c), (b,b) } be a relation onA, then the symmetric closure (对称闭包) ofRis ________________________.
A . [0] B. [2] C. [3] D. [1]
三、计算题(每小题6分，共30分)
1. Solve therecurrence relation (求解递推关系) cn=-8cn-1-16cn-2with initial conditionsc1=-1 andc2=8.
2.Letf:ZZbe a function given by . Isfone to one or onto? Explain.
11. Use the Huffman code tree in Figure 3 to decode the message: 000110000101 (用图3中的Huffman编码树，译出信息: 000110000101) ___________.
12. Whether the graph shown in Figure 4 has an Euler circuit? (图4中是否具有欧拉回路？).
3.LetS={1, 2, 3, 6, 12},a*bis defined as GCD (a,b). Determine whether (S, *) is a semigroup, a monoid. If it is a monoid, specify the identity.
山东建筑大学试卷共5页第4页
二选择题每小题2分共20分在每小题列出的四个选项中只有一个选项符合题目的要求请将其代码填写在题后的括号内
山东建筑大学试卷共5页第1页
2012至2013学年第1学期考试时间：120分钟
课程名称：离散数学AB卷考试形式：闭卷
年级：11专业：信息与计算科学；层次：本科
题号
一
二
三
四
五
总分
分数
一、填空题（每小题2分，共30分）
5.LetA= {3, 7, 12}, define a relationRonAbyxRyif and only ifx-y<4.LetMR=(mij) be the matrix of the relationR, thenm21=,m22=.
6.Determine whether the lattice shown in Figure 1 is distributive (分配的). _______.
1.LetA={a,b,c,d, e, f} andRbe a relation onAwhose matrix is .
(a)Prove tRis an equivalence relation.
(b)Give the partition ofAcorresponding toR.
山东建筑大学试卷共5页第5页
···········································································································装订线··································································································
4.Let (D30,) be the lattice of all positive divisors of 30 andxymeansx|y.
(a) Draw the Hasse diagram of the lattice.
(b)Determine whether (D30,) is a Boolean algebra. Explain.
e Kruskal’s algorithm to find a minimal spanning tree for the graph in Figure 7. List the edges in the order in which they are chosen.
四、证明题（每小题7分，共14分）
15.LetA= {Ø, a, {a, c}} be a set.Determine whether “{a,c}A” is true or false ._______.
二、选择题(每小题2分，共20分)在每小题列出的四个选项中只有一个选项符合题目的要求，请将其代码填写在题后的括号内。
1.Which of the following statements is the negation of “6is even or -5is
2. LetP(x) be the statement "xcan speak Russian" and letQ(x) be the statement "xknows the computer language C++." Express the statement “There is a student at your school who can speak Russian and who knows C++” in terms ofP(x),Q(x), quantifiers(量词), and logical connectives ___________. The domain for quantifiers consists of all students at your school.
1. Letpbe the proposition: "The message is scanned for viruses" andq: "The message was sent from an unknown system”. Write the following statement in terms ofp,qand logical connectives. "It is necessary to scan the message for viruses whenever it was sent from an unknown system."___________
A .Q(1) B . (x) Q (x) C. (x)Q(x) D. Q(-1)
6. LetR={ (a, a), (b, b), (c, c)} be a relation on a set A={a, b, c}, which is the best forR? (哪一项最适合R) ( )
A. R is an equivalence relation. B.Ris reflexive
C.Ris symmetric D.Ris transitive
7. Which of the followingHassediagrams represents a Boolean algebra? ( )
8.Let G be the graph shown in Figure 6.How many edges need to be removed to produce a spanning tree in G .(在图6中,需要删除多少条边才可能产生一棵生成树) ? ( )
A .p=1,q=1,r=0 B.p=0,q=1,r=1
C.p=1,q=0,r=1 D.p=0,q=0,r=1
3. In the following, which is true? ( )
A. {}{}={} B. {}{0, 1,3} C.{0, 1} D.{0}{0, 1,3}
4.LetA=Z,defined a relationRonAbyaRbif and only if |a-b|=3, which is true forR? ( )
2. LetGbe the set of all nonzero real numbers and let .
Show that (G,) is a group.
五、应用题（6分）
若传递c, d, g, k, l, o, u的频率分别为c: 5%，d: 3%，g: 8%，k: 2%，l: 6%，o: 12%，u: 15%，求传输它们的最佳前缀码。给出good luck的编码信息。
7. In a poset whose Hasse diagram shown in Figure 2, the least upper bound (最小上界) of B={ c, d, e} is.
8. Letnbe a positive integer and letDnbe the set of all positive divisors of n. Then Dnis a lattice under the relation of divisibility. Determine whether D60is a Boolean algebra. _________
山东建筑大学试卷共5页第2页
13.LetS={a,b}, for P(S), *is defined as intersection, then (P(S), *) is a monoid, and the identity is_____________.
14. If S is a nonempty set,for A, BP(S), *is defined asA*B=AB (symmetric difference), then (P(S), *) is a group, and its identity is_________.
A.6is even or –5 is not negative. B.6is odd and –5 is not negative.
C.6is odd or –5 is not negative. D.6is even and –5 is negative.
2. When the propositionp(qr) is true? ( )
A.Ris antisymmetric B.Ris reflexive
C.Ris symmetric D.Ris asymmetric
5. Let Q(x) be the statement “x+1<3”. If the universe of discourse consists of the integers, in the following, which is false? ( )
A . 0 B. 1 C. 2 D. 3
9.In the Boolean algebra D30, the complement (补元) of 3 is ( )
A. 10 B.6 C. 15 D. 30
山东建筑大学试卷共5页第3页
10. Let Z4be a group with operationas the Table 1, which is the inverse of [3]? ( )
9. Is the poset A={2, 3, 6, 18} under the relation of divisibility a lattice? _______.
10.Supposef:NNhas the rulef(ቤተ መጻሕፍቲ ባይዱ)=3n+2. Determine whetherfis one-to-one? ___________.
3.LetA={1，2，…，30}, define a relationRonAbyaRbif and only if |a-b|5(a,bA), determine whetherRis symmetric.
4. LetA={a,b,c},R={ (b,c), (b,b) } be a relation onA, then the symmetric closure (对称闭包) ofRis ________________________.
A . [0] B. [2] C. [3] D. [1]
三、计算题(每小题6分，共30分)
1. Solve therecurrence relation (求解递推关系) cn=-8cn-1-16cn-2with initial conditionsc1=-1 andc2=8.
2.Letf:ZZbe a function given by . Isfone to one or onto? Explain.
11. Use the Huffman code tree in Figure 3 to decode the message: 000110000101 (用图3中的Huffman编码树，译出信息: 000110000101) ___________.
12. Whether the graph shown in Figure 4 has an Euler circuit? (图4中是否具有欧拉回路？).
3.LetS={1, 2, 3, 6, 12},a*bis defined as GCD (a,b). Determine whether (S, *) is a semigroup, a monoid. If it is a monoid, specify the identity.
山东建筑大学试卷共5页第4页
二选择题每小题2分共20分在每小题列出的四个选项中只有一个选项符合题目的要求请将其代码填写在题后的括号内
山东建筑大学试卷共5页第1页
2012至2013学年第1学期考试时间：120分钟
课程名称：离散数学AB卷考试形式：闭卷
年级：11专业：信息与计算科学；层次：本科
题号
一
二
三
四
五
总分
分数
一、填空题（每小题2分，共30分）
5.LetA= {3, 7, 12}, define a relationRonAbyxRyif and only ifx-y<4.LetMR=(mij) be the matrix of the relationR, thenm21=,m22=.
6.Determine whether the lattice shown in Figure 1 is distributive (分配的). _______.
1.LetA={a,b,c,d, e, f} andRbe a relation onAwhose matrix is .
(a)Prove tRis an equivalence relation.
(b)Give the partition ofAcorresponding toR.
山东建筑大学试卷共5页第5页
···········································································································装订线··································································································
4.Let (D30,) be the lattice of all positive divisors of 30 andxymeansx|y.
(a) Draw the Hasse diagram of the lattice.
(b)Determine whether (D30,) is a Boolean algebra. Explain.
e Kruskal’s algorithm to find a minimal spanning tree for the graph in Figure 7. List the edges in the order in which they are chosen.
四、证明题（每小题7分，共14分）
15.LetA= {Ø, a, {a, c}} be a set.Determine whether “{a,c}A” is true or false ._______.
二、选择题(每小题2分，共20分)在每小题列出的四个选项中只有一个选项符合题目的要求，请将其代码填写在题后的括号内。
1.Which of the following statements is the negation of “6is even or -5is
2. LetP(x) be the statement "xcan speak Russian" and letQ(x) be the statement "xknows the computer language C++." Express the statement “There is a student at your school who can speak Russian and who knows C++” in terms ofP(x),Q(x), quantifiers(量词), and logical connectives ___________. The domain for quantifiers consists of all students at your school.
1. Letpbe the proposition: "The message is scanned for viruses" andq: "The message was sent from an unknown system”. Write the following statement in terms ofp,qand logical connectives. "It is necessary to scan the message for viruses whenever it was sent from an unknown system."___________
A .Q(1) B . (x) Q (x) C. (x)Q(x) D. Q(-1)
6. LetR={ (a, a), (b, b), (c, c)} be a relation on a set A={a, b, c}, which is the best forR? (哪一项最适合R) ( )
A. R is an equivalence relation. B.Ris reflexive
C.Ris symmetric D.Ris transitive
7. Which of the followingHassediagrams represents a Boolean algebra? ( )
8.Let G be the graph shown in Figure 6.How many edges need to be removed to produce a spanning tree in G .(在图6中,需要删除多少条边才可能产生一棵生成树) ? ( )
A .p=1,q=1,r=0 B.p=0,q=1,r=1
C.p=1,q=0,r=1 D.p=0,q=0,r=1
3. In the following, which is true? ( )
A. {}{}={} B. {}{0, 1,3} C.{0, 1} D.{0}{0, 1,3}
4.LetA=Z,defined a relationRonAbyaRbif and only if |a-b|=3, which is true forR? ( )
2. LetGbe the set of all nonzero real numbers and let .
Show that (G,) is a group.
五、应用题（6分）
若传递c, d, g, k, l, o, u的频率分别为c: 5%，d: 3%，g: 8%，k: 2%，l: 6%，o: 12%，u: 15%，求传输它们的最佳前缀码。给出good luck的编码信息。
7. In a poset whose Hasse diagram shown in Figure 2, the least upper bound (最小上界) of B={ c, d, e} is.
8. Letnbe a positive integer and letDnbe the set of all positive divisors of n. Then Dnis a lattice under the relation of divisibility. Determine whether D60is a Boolean algebra. _________
山东建筑大学试卷共5页第2页
13.LetS={a,b}, for P(S), *is defined as intersection, then (P(S), *) is a monoid, and the identity is_____________.
14. If S is a nonempty set,for A, BP(S), *is defined asA*B=AB (symmetric difference), then (P(S), *) is a group, and its identity is_________.