Discrete Mathematics(University of Science and Technology Beijing) 中国大学mooc慕课答案2024版 m62882

1. Propositional logic of mathematical logic Chapter 1 unit tests

1、 Which is a proposition in the following statements?

答案: The sun is a planet.

2、 Which is an atomic statement in the following statements?

答案: Zhang San and LiSi are friends.

3、 Suppose P: I’m ChineseQ: I come from Beijing”Unless I am from Beijing, I am not Chinese” is the result of the symbolic statement:

答案:

4、 Which is a false statement in the following statements?

答案: 2 is a prime number only if USTB is not in Beijing. 

5、 Which is the simplified form of the following combination formulas:?

答案:

6、 According to the implication law, which one is  equivalented in the following formulas？

答案:

7、 German Nazi leader goebbels once said that a lie repeated three times becomes a truth.Which law does this violate?

答案: Idempotent law

8、 “Stinky tofu stinks and smells good” is a false statement.Which law does this use?

答案: Law of contradiction

9、 Our cat either sleeps on the balcony or eats meat in the kitchen.I haven’t found it on the balcony yet, so it must be in the kitchen.What is the law of inference used?

答案: Disjunctive syllogism

10、 What is the dual of?

答案:

11、 What is the type of?

答案: Nontautological satisfiable formula

12、 Which one is conjunctive and disjunctive paradigms?

答案:

13、 Which are propositions in the following statements?(multiple choices)

答案: The shot is not a ball. ;
If he doesn’t play, we’ll lose.;
Liu xiang took less than 13 seconds in the 110m hurdles. He is a real athlete.

14、 Which are not compound propositions in the following propositions? Multiple choices

答案: I have classes every day. ;
Are there people on Mars?

15、 Which is a false statement in the following statements?

答案: 2 is a prime number only if USTB is not in Beijing. 

2. Predicate logic of mathematical logic Chapter 2 unit tests

1、 Which one is not a predicate formula?

答案:

2、 In the formula，the free variable is()

答案: y

3、 Which one is wrong?

答案:

4、 Which one is wrong?

答案:

5、 Which one is wrong?

答案:

6、 There is a proposition as follows: any real number x, there will always be real number y, which is less than x. Set: F(x): x is a real number. G(x, y) : x < y What is the symbolic representation of the correctness of the proposition in the scope of predicate logic?

答案:

7、 Which one is the prenex normal form?

答案:

8、 Suppose the individual domain is a set of integers. Which has a true value of 0 in the following formulas? 

答案: ;
;

9、 Which is correct in the following formula?

答案: ;

10、 Which is not the Negation of formula?

答案: ;
;

11、 The type of well-formed formulais Tautology

答案: 错误

12、 Set the individual domain {1,2}, the predicate P (1) = 1, P (2) = 0, Q (1) = 0, Q (2) = 1,so the truth-value ofis 1.

答案: 正确

13、 The type of formulais Tautology 

答案: 正确

14、 Let the individual domain be A = {a, b}, The formulaafter eliminating the quantifier should be.

答案: 错误

15、 Determine whether this formula is correct:

答案: 错误

16、 Which is not the Negation of formula?

答案: ;
;

17、 Set the individual domain {1,2}, the predicate P (1) = 1, P (2) = 0, Q (1) = 0, Q (2) = 1,so the truth-value ofis 1.

答案: 正确

18、 The type of formulais Tautology 

答案: 正确

3. Set of set theory Chapter 3 unit tests

1、 The cardinal numbers of which set is three?

答案:

2、 What’s wrong?

答案:

3、 Which of the following is wrong ?

答案:

4、 On [1, 100], how many integers can’t be divided by 3 and 5 or 7?

答案: 45

5、 A. B and C are arbitrary sets. What’s right?

答案:

6、 A. B is any set. What’s the error?

答案:

7、 A. B and C are arbitrary sets. What’s right?

答案:

8、 Known ,what the method of solving the order of the value of  and ?

答案: ;

9、 Due to ,what can be deduced?

答案: ;

10、 What laws are used in the following reasoning? 

答案: Law of identity;
Law of distribution;
Law of zeros

11、

答案: 正确

12、 Suppose a, B and C are any set,

答案: 错误

13、

答案: 错误

14、

答案: 正确

15、

答案: 正确

16、 Suppose a, B and C are non null sets,ifand,then.

答案: 错误

17、 A. B and C are arbitrary sets. What’s right?

答案:

18、 What laws are used in the following reasoning?

答案: Law of identity;
Law of distribution;
Law of zeros

19、 Suppose a, B and C are any set,

答案: 错误

4.Binary relation of set theory unit test of binary relation

1、 Which of the follwing has the Transitive relationship?

答案: The same year relationship of ancient imperial examination

2、 Which of the following is equivalent?

答案: Logical equivalence of propositional formula

3、 A. B, C and D are arbitrary sets. What’s right?

答案:

4、 What’s right?

答案: The poset of a finite set must have the Maximal element

5、 R is a binary relationship, and , which of the following is not necessarily transitive?

答案: ;
;

6、 There are 255 binary relations on a = {1, 2, 3, 4, 5}

答案: 错误

7、 A = {a, b}, R is the relationship on A, r = {< a, a >, < b, b >}, then R is not transitive.

答案: 错误

5. Functions of set theory functional unit test

1、 Considering the function on the real number set, the analytic expression of is

答案:

2、 Which of the following relationships can form a function?

答案: f={<x, y>| x,y∈N, and Y is the number of prime numbers less than x}

3、 Which of the following functions has an inverse function?

答案: f:N->N, f(x)=x

4、 Suppose A and B are finite sets,|A|=n，|B|=m，and n,m>0，then 

答案:

5、 Which of the following statement is correct ()

答案: A and B are two equal sets. The function from A to B is unijection only if it is a hyperjection.;
The inverse relations of functions are not always functions.;
Only bijective function is invertible, and its inverse function is bijective.

6、 The number of elements in set A is n, and the number of elements in set B is m. The different surjections from set A to set B are equal to the number of m-containing elements in set A of n elements.

答案: 正确

7、 The number of elements of set a is n, and the number of elements of set B is m. There are m*n kinds of different bijections from set a to set B.

答案: 错误

8、 Considering the function on the real number set, the analytic expression of is

答案:

9、 Suppose A and B are finite sets,|A|=n，|B|=m，and n,m>0，then 

答案:

7.Algebraic system of algebraic structure unit test of algebra system

1、 Let G be a non-zero real number set R * for the algebraic system composed of ordinary multiplication, which of the following functions is the endomorphism of G?

答案: f(x) = |x|

2、 Which of the following is not a binary operation on the integer set Z?

答案: division

3、 On the natural number set N, which of the following operations can be combined?

答案: a*b=max{a,b}

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,

1. Propositional logic of mathematical logic Chapter 1 unit tests

1、 Which is a proposition in the following statements?

答案: The sun is a planet.

2、 Which is an atomic statement in the following statements?

答案: Zhang San and LiSi are friends.

3、 Suppose P: I’m ChineseQ: I come from Beijing”Unless I am from Beijing, I am not Chinese” is the result of the symbolic statement:

答案:

4、 Which is a false statement in the following statements?

答案: 2 is a prime number only if USTB is not in Beijing. 

5、 Which is the simplified form of the following combination formulas:?

答案:

6、 According to the implication law, which one is  equivalented in the following formulas？

答案:

7、 German Nazi leader goebbels once said that a lie repeated three times becomes a truth.Which law does this violate?

答案: Idempotent law

8、 “Stinky tofu stinks and smells good” is a false statement.Which law does this use?

答案: Law of contradiction

9、 Our cat either sleeps on the balcony or eats meat in the kitchen.I haven’t found it on the balcony yet, so it must be in the kitchen.What is the law of inference used?

答案: Disjunctive syllogism

10、 What is the dual of?

答案:

11、 What is the type of?

答案: Nontautological satisfiable formula

12、 Which one is conjunctive and disjunctive paradigms?

答案:

13、 Which are propositions in the following statements?(multiple choices)

答案: The shot is not a ball. ;
If he doesn’t play, we’ll lose.;
Liu xiang took less than 13 seconds in the 110m hurdles. He is a real athlete.

14、 Which are not compound propositions in the following propositions? Multiple choices

答案: I have classes every day. ;
Are there people on Mars?

15、 Which is a false statement in the following statements?

答案: 2 is a prime number only if USTB is not in Beijing. 

2. Predicate logic of mathematical logic Chapter 2 unit tests

1、 Which one is not a predicate formula?

答案:

2、 In the formula，the free variable is()

答案: y

3、 Which one is wrong?

答案:

4、 Which one is wrong?

答案:

5、 Which one is wrong?

答案:

6、 There is a proposition as follows: any real number x, there will always be real number y, which is less than x. Set: F(x): x is a real number. G(x, y) : x < y What is the symbolic representation of the correctness of the proposition in the scope of predicate logic?

答案:

7、 Which one is the prenex normal form?

答案:

8、 Suppose the individual domain is a set of integers. Which has a true value of 0 in the following formulas? 

答案: ;
;

9、 Which is correct in the following formula?

答案: ;

10、 Which is not the Negation of formula?

答案: ;
;

11、 The type of well-formed formulais Tautology

答案: 错误

12、 Set the individual domain {1,2}, the predicate P (1) = 1, P (2) = 0, Q (1) = 0, Q (2) = 1,so the truth-value ofis 1.

答案: 正确

13、 The type of formulais Tautology 

答案: 正确

14、 Let the individual domain be A = {a, b}, The formulaafter eliminating the quantifier should be.

答案: 错误

15、 Determine whether this formula is correct:

答案: 错误

16、 Which is not the Negation of formula?

答案: ;
;

17、 Set the individual domain {1,2}, the predicate P (1) = 1, P (2) = 0, Q (1) = 0, Q (2) = 1,so the truth-value ofis 1.

答案: 正确

18、 The type of formulais Tautology 

答案: 正确

3. Set of set theory Chapter 3 unit tests

1、 The cardinal numbers of which set is three?

答案:

2、 What’s wrong?

答案:

3、 Which of the following is wrong ?

答案:

4、 On [1, 100], how many integers can’t be divided by 3 and 5 or 7?

答案: 45

5、 A. B and C are arbitrary sets. What’s right?

答案:

6、 A. B is any set. What’s the error?

答案:

7、 A. B and C are arbitrary sets. What’s right?

答案:

8、 Known ,what the method of solving the order of the value of  and ?

答案: ;

9、 Due to ,what can be deduced?

答案: ;

10、 What laws are used in the following reasoning? 

答案: Law of identity;
Law of distribution;
Law of zeros

11、

答案: 正确

12、 Suppose a, B and C are any set,

答案: 错误

13、

答案: 错误

14、

答案: 正确

15、

答案: 正确

16、 Suppose a, B and C are non null sets,ifand,then.

答案: 错误

17、 A. B and C are arbitrary sets. What’s right?

答案:

18、 What laws are used in the following reasoning?

答案: Law of identity;
Law of distribution;
Law of zeros

19、 Suppose a, B and C are any set,

答案: 错误

4.Binary relation of set theory unit test of binary relation

1、 Which of the follwing has the Transitive relationship?

答案: The same year relationship of ancient imperial examination

2、 Which of the following is equivalent?

答案: Logical equivalence of propositional formula

3、 A. B, C and D are arbitrary sets. What’s right?

答案:

4、 What’s right?

答案: The poset of a finite set must have the Maximal element

5、 R is a binary relationship, and , which of the following is not necessarily transitive?

答案: ;
;

6、 There are 255 binary relations on a = {1, 2, 3, 4, 5}

答案: 错误

7、 A = {a, b}, R is the relationship on A, r = {< a, a >, < b, b >}, then R is not transitive.

答案: 错误

5. Functions of set theory functional unit test

1、 Considering the function on the real number set, the analytic expression of is

答案:

2、 Which of the following relationships can form a function?

答案: f={<x, y>| x,y∈N, and Y is the number of prime numbers less than x}

3、 Which of the following functions has an inverse function?

答案: f:N->N, f(x)=x

4、 Suppose A and B are finite sets,|A|=n，|B|=m，and n,m>0，then 

答案:

5、 Which of the following statement is correct ()

答案: A and B are two equal sets. The function from A to B is unijection only if it is a hyperjection.;
The inverse relations of functions are not always functions.;
Only bijective function is invertible, and its inverse function is bijective.

6、 The number of elements in set A is n, and the number of elements in set B is m. The different surjections from set A to set B are equal to the number of m-containing elements in set A of n elements.

答案: 正确

7、 The number of elements of set a is n, and the number of elements of set B is m. There are m*n kinds of different bijections from set a to set B.

答案: 错误

8、 Considering the function on the real number set, the analytic expression of is

答案:

9、 Suppose A and B are finite sets,|A|=n，|B|=m，and n,m>0，then 

答案:

7.Algebraic system of algebraic structure unit test of algebra system

1、 Let G be a non-zero real number set R * for the algebraic system composed of ordinary multiplication, which of the following functions is the endomorphism of G?

答案: f(x) = |x|

2、 Which of the following is not a binary operation on the integer set Z?

答案: division

3、 On the natural number set N, which of the following operations can be combined?

答案: a*b=max{a,b}

,

4、 Let A = {2, 5, 8}, the binary operation * is defined as: a * b = max {a, B}, then the unit element in < A, * > is

答案: 2

5、 Let A = {2,5,8}, the binary operation * is defined as: a * b = max {a, b}, then the zero element in < A, * > is

答案: 8

6、 Let A = {2, 5, 8}, the binary operation * is defined as: a * b = min {a, b}, then in < A, * > the unit element is

答案: 8

7、 Let A = {2, 5, 8}, the binary operation * is defined as: a * b = min {a, b}, then in the unique point < A, * > the zero element is

答案: 2

8、 Q is the rational number set N, and the definition operation * on Q is a*b= a + b – ab, then the unit elements of <Q, * > is

答案: 0

9、 Let v = < Z, + > and which of the following functions is the endomorphism on V?

答案: f(x)=5x

10、 Let G1=<{0,1,2},∘>，G2=<{0.1},*>,where ∘denotes modulo 3  addition, * denotes modulo 2 multiplication, then the unit element of product algebra 

答案: <0,1>

11、 The following sets are subsets of N, which of the sets are closed under normal addition operations

答案: {x | X is a multiple of 30}

12、 On the natural number set n, which of the following operations are commutative?

答案: ab=max{a,b};
ab=|a-b|

13、 Which of the following is a binary operation on n?

答案: addition;
multiplication

14、 Which of the following is not a binary operation on n?

答案: subtraction;
division

15、 V=<R,×>, which of the following functions are homomorphisms of V to v?

答案: f(x)=x*x;
f(x)=1/x

16、 In the operation table, if the element arrangement order of the row and column where an element is located is consistent with that of the header element, then this element is the unit element.

答案: 正确

17、 In an operation table, if the elements of an element’s row and column are the element itself, then the element is zero.

答案: 正确

18、 If the elements of the operation table are symmetrically distributed about the main diagonal, the operation can be combined.

答案: 错误

19、 Let G1=<{0,1,2},∘>，G2=<{0.1},*>,where ∘denotes modulo 3  addition, * denotes modulo 2 multiplication, then the unit element of product algebra 

答案: <0,1>

8. A Preliminary Study of Group Theory of Algebraic Structure preliminary unit test of group theory

1、 Any subgroup of a finite group of order 6 must not be()

答案: Rank 4

2、 Let a be the generator of group of order 10, then the fourth power of a is the element of order ().

答案: Rank 5

3、 Let a be the generator of group of order 10, then the third power of a is the element of order ().

答案: Rank 10

4、 Let a be the generator of group of order 12, then the second power of a is the element of order ().

答案: Rank 6

5、 Let a be the generator of group of order 12, then the third power of a is the element of order ().

答案: Rank 4

6、 If <G,*> is a group, then  * is()

答案: Have identity element and can be combined

7、 The order of a subgroup of a sixth order group can be

答案: 2,3

8、 Let K = {e , a , b , c}, < K, * > is a Klein quaternion group, then the inverse element of element a is

答案: a

9、 The subgroups of group < Z4, ⊕ > is

答案: <{0,2},⊕>

10、 Which of the following in algebraic systems <S,> is a group?

答案: S={1，3，4，5，9}， is module 11 multiplication

11、 If S={0，1}, * is a normal multiplication, then < S , * > is

答案: It’s just a monoid , but not a group

12、 Binary operation * defined on rational set Q.If any rational number x, y has x * y = x + y-xy, then q satisfies

答案: If any rational number x, x is not equal to 1, there is inverse element

13、 Which of the following are generators of cyclic group < a > of order 8 ?

答案: a;
The third power of a;
The fifth power of a;
a的7次幂

14、 Which of the following are subgroups of cyclic group <a> of order 8?

答案: {e};
<a>;
Groups generated by the power of 4 of a;
Groups generated by the power of 2 of a

15、 Which of the following are nontrivial subgroups of cyclic group < a > of order 8?

答案: Groups generated by the power of 4 of a;
Groups generated by the power of 2 of a

16、 If <G,*> is a group, then * is()

答案: Satisfied combination law;
Have Unit element;
Every unit has inverse element

17、 In a group 〈G,*〉, if the order of element a in G is k, then the order of the inverse element of a is -k.

答案: 错误

18、 Groups of prime order must be cyclic groups.

答案: 正确

19、 The number of elements of order 2 in even order groups must be odd

答案: 正确

20、 The number of elements with order greater than 2 in a finite group must be even.

答案: 正确

21、 There is no zero element in a group with at least two elements.

答案: 正确

22、 a is the generator of a group〈G，〉, then the inverse of a is also the generator of a group〈G，〉.

答案: 正确

23、 There must be an element of order 2 in an even order group.

答案: 正确

24、 Group G has no idempotent except unit element.

答案: 正确

10. The basic concept of graph in graph theory basic concept unit test of Graphs

1、 If G is a self complementary graph of order n, what the edge numbers of G?

答案: n(n-1) / 4

2、 If undirected graph G has 16 edges, 3 4-degree nodes, 4 3-degree nodes, and the degrees of other vertices are less than or equal to 2, how many vertices are there at least in G?

答案: 11

3、 Now there are n boxes. If there is exactly one ball of the same color in each of the two boxes, and there are exactly two balls of each color, and they are put in different boxes, how many different colors are the balls in these n boxes?

答案: n(n-1) / 2

4、 What’s right?

答案: Graph isomorphism is an equivalent relation

5、 What’s right?

答案: In any graph, the sum of degrees of vertices is equal to twice the number of edges.;
The necessary conditions for isomorphism of two graphs are equal order, equal edge number and equal degree sequence.;
A graph with a given point or edge weight is called a weighted graph.;
A graph with specified symbols for both vertices and edges is called a calibration graph.

6、 In a graph, there are both directed and undirected edges. Such a graph is called a mixed graph. Is that right?

答案: 正确

7、 A simple graph in which all vertices have the same degree is called a regular graph. Is that right?

答案: 正确

11. Connectivity of Graph Theory Graphs connectivity unit test of Graphs

1、 In graph G, the distance D (U, V) from vertex u to V does not satisfy which of the following properties?

答案: d(u, u) = ∞

2、 In undirected graph G = < V, E > is the connected relation between vertices on V?

答案: equivalence relation

3、  What is wrong with the following statement?

答案: If an edge in an undirected graph is a cut edge, it must be included in any simple circuit in the graph.

4、 What is the number of edges in a connected graph with n vertices?

答案: At least n-1.

5、 which of the following statements are true?

答案: If all the vertices in a path are different, the path is called the basic path.;
If a vertex of a connected undirected graph is a cut point of the graph, it must be a joint point of a pair of vertices.;
If any pair of vertices are reachable to each other, then this graph is a strongly connected graph.

6、 which of the following statements are true?

答案: A digraph G is strongly connecte only if there is a loop in G, it passes through every vertex at least once.;
A directed graph is a one-way connected graph if and only if it has a path through all nodes.;
In G=<V,E> with n vertices, if there is a path from u to V, there must be a path shorter than n-1 from u to v.

7、 If there is a path between two vertices, the path between the two vertices is still obtained by deleting part of the path between repeated vertices on the path, and the length is less than the number of vertices in the graph. Is this statement correct?

答案: 正确

8、 Let n be the number of vertices of a connected simple undirected graph, and K be the minimum degree of vertices in the graph. Then, if there must be a basic path with a length of 2K in the graph?

答案: 正确

12. Matrix Representation of Graph Theory Graphs graph theory and matrix representation unit test of

1、 It is known that the adjacency matrix of a digraph D is as follows. The nodes corresponding to the matrix are v1 to v4 from left to right. How many paths are there in D with the length of 3 from v1 to v4?

答案: 2

2、 The reachability matrix of a digraph is known as follows,the graph is a (  )?

答案: Unilateral connected graph 

3、 Given that the correlation matrix of a simple digraph is as follows, which of the following pairs of nodes are reachable to each other?

答案: v3, v4

4、 What is wrong with the following statement?

答案: A digraph is a unilateral connected graph only if all elements of the reachable matrix are 1.

5、 Which of the following statements are true?

答案: If a connected graph has r vertices, the rank of its complete correlation matrix is r-1.;
The sum of the elements of each row of the completely related matrix corresponds to the degree of the vertex.;
The elements in a row of the complete incidence matrix are all 0, and the corresponding vertices are isolated points.;
The two columns corresponding to two parallel sides in the complete incidence matrix are the same.

6、 Which of the following statements are true?

答案: The diagonals of adjacency matrix are all zero only if the graph has no rings.;
The adjacency matrix of a digraph is related to the order of the elements in its vertex set.;
In the adjacency matrix of a digraph, a digraph must be constructed according to the n-matrix of given elements 0 and 1.

7、 In adjacency matrix, exchange rows must exchange columns . Is that right?

答案: 正确

8、 It is known that the adjacency matrix of a digraph D is as follows. The nodes corresponding to the matrix are v1 to v4 from left to right. How many paths are there in D with the length of 3 from v1 to v4?

答案: 2

9、 The reachability matrix of a digraph is known as follows,the graph is a (  )?

答案: Unilateral connected graph 

10、 Given that the correlation matrix of a simple digraph is as follows, which of the following pairs of nodes are reachable to each other?

答案: v3, v4

13. Special Diagrams Unit test of special drawing

1、 A connected nontrivial undirected graph G has an Euler loop only if G is ()

答案: No singularity node

2、 Between a cut edge set and any spanning tree()

答案: At least one common edge

3、 If a tree has 7 leaves, 3 3 degree nodes and the rest are all 4 degree nodes, then the tree has () 4 degree nodes.

答案: 1

4、 Which of the following is the prefix encoding

答案: 00,10,110,011

5、 Which of the following statements are true?

答案: A graph with a circuit that passes through every edge of the graph once and only once is called an Euler graph.;
An undirected graph has an Euler path only if the graph is connected and has two or zero odd degree vertices.;
Undirected graphs have an Euler loop if and only if the graph is connected and all vertex degrees are even.;
A directed graph has a one-way Euler path, only if the graph is connected, and except for two vertices, the degree of entry of each vertex is equal to the degree of exit.

6、 Which of the following statements are true?

答案: If G is a simple graph with n vertices, if the sum of degrees of each pair of vertices in G is greater than or equal to n-1, then there is a Hamiltonian path in G.;
If G is a simple undirected graph, G is a Hamiltonian graph and only if its closure is a Hamiltonian graph.;
Undirected graph G is bipartite only if the length of all circuits in G is even.;
A connected undirected graph has at least one spanning tree.

7、 If any two vertices of a acyclic graph are connected by a unique path, then this graph is a tree. Is that right?

答案: 正确

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