WorksheetsIA-2 section-19
Total questions: 42
Worksheet time: 21mins
Using a Cayley Table, how can we identify if a group is Abelian?
a. If there are no duplication of elements in all rows and columns.
b. If the group order is even.
c. If the elements in the table are symmetric with respect to the main diagonal.
Which of the following is a property of a group?
Every element has a unique inverse.
The group operation is commutative.
There must be finite elements
There can be at most 2 identity elements
Which of the following sets, with the operation of addition, forms a subgroup of the group of integers Z ?
The set of all even integers.
The set of all odd integers.
The set of all positive integers.
The set of all prime numbers.
The set of all real numbers under the usual multiplication operation is not a group since
multiplication is not a binary operation
multiplication is not associative
identity element does not exist
zero has no inverse
Which property states that the operation in a group must produce an element that is also in the group?
Associativity
Closure
Inverse Element
Identity Element
Which operation is associative?
Division of real numbers
Subtraction of integers
None of the above
Addition of integers
Which of the following is not a commutative operation?
Multiplication of real numbers
Subtraction of integers
Multiplication of integers
Addition of real numbers
What is a group in algebraic structures?
A set with a binary operation that satisfies closure, associativity, identity, and inverses
A set with only one element
A set with no operations defined
A set that only contains integers
Which operation is associative?
Division of real numbers
Subtraction of integers
None of the above
Addition of integers
What is the order of an element in a group?
The number of operations defined on the group
The number of elements in the group
The smallest positive integer n such that g^n = e
The number of inverses of the element
Which of the following is true about the set of rational numbers under addition?
It has no identity element
It has an identity element of 0
It is not closed under addition
It has no inverses
What is a semigroup?
A set with closure and associativity properties
A set with only one element
A set that is not closed
A set that has an identity element
Which of the following is an example of a monoid?
Rational numbers under division
Integers under subtraction
Natural numbers under multiplication
Positive integers under addition
Consider a bipartite graph G=(U,V,E) . Which of the following statements is true?
U and V must have the same number of vertices
Every edge in E connects a vertex in U to a vertex in V
G must be a tree
G must be a complete graph
Which of the following statements is true for a bipartite graph?
It can have self-loops
It can have multiple edges between the same pair of vertices
It can be disconnected
It must be a complete graph
In a bipartite graph, if one set has 5 vertices and the other set has 7 vertices, what is the maximum number of edges the graph can have?
12
35
25
30
Which set represents the vertices of the graph?
{1,2}, {2,3}. {2,4}, {4,5}, {4,6}
{1, 2, 3, 4, 5, 6,}
{1, 2, 3, 4, 5, 6, 7}
{1,2}, {2,3}. {2,4}, {4,5}, {4,6}, {6,7}
What is the degree of vertex 4?
3
4
5
7
Which two vertices are adjacent vertices?
5 is adjacent to 6
3 is adjacent to 6
4 is adjacent to 1
3 is adjacent to 2
The graph is an example of a
Path
Cycle
Complete Bipartite Graph
Complete Graph
Which describes the edges of the graph?
{A,B}, {A,C}, {A, E}, {B,C}, {B,E}
A, B, C, D, E
{A,B}, {A,C}, {A, E}, {B,C}, {B,E}, {C,D}
{A,B}, {A,C}, {A, E}
Which notation is correct for the complete bipartite graph?
K2,3
K3
K2
K1, 3
What is the number of edges present in a complete graph having n vertices?
n
n-1
n(n-1)/2
n+1
Which of the following is true?
A graph may contain no edges and many vertices
A graph may contain many edges and no vertices
A graph may contain no edges and no vertices
A graph may contain no vertices and many edges
The given Graph is regular.
True
False
A graph G is r-Regular if,
deg(v)=r for all v in V(G)
d(u,v)=r for all u,v in V(G)
|V(G)|=r
|E(G)|=r
Which of the following statements is/are true?
Number of odd degree vertices is even.
Sum of degrees of all vertices is even.
Both A and B
Neither A nor B
Which of the following is not a property of Complete Bipartite Graph?
It consists of two sets of vertices X and Y.
The vertices of set X join only with the vertices of set Y.
The vertices within the same set do not join.
Each vertex in the same set are adjacent.
The degree of a vertex is the number of edges incident with that vertex.
True
False
Units of Z are
0 and 1
1 and -1
-1 and 0
-1, 0 and 1
Which is an example of a disconnected graph?
None are disconnected graphs
What type of graph is this?
Complex
Complement
Complete
Complete Bipartite
Which vertex has a loop?
(a)
What is the number of edges present in a complete graph having n vertices?
2n(n+1)
2n(n−1)
n
Information given is insufficient
What will be the number of edges in a complete bipartite graph Km,n
m+n
m×n
Data Insufficient
Which graph is not a complete graph?
Determine the sum of degrees.
16
8
4
13
Euler's handshaking lemma states that the sum of the degrees of the vertices is equal to....
twice the number of vertices
twice the number of edges
half the number of vertices
half the number of edges
A simple graph ...
has no edges
no loops
no multiple edges
no loops nor multiple edges
A vertex of degree one is called as
Null vertex
Singlet vertex
Pendant Vertex
Individual Vertex
Which of the following statements is/are true?
Number of odd degree vertices is even.
Sum of degrees of all vertices is even.
Both A and B
Neither A nor B
Which of the following statements is/are true?
Number of odd degree vertices is even.
Sum of degrees of all vertices is even.
Both A and B
Neither A nor B
