Master Discrete Mathematics Concepts: Online Quizzes and Learning Resources

Question 1

In a group of 101 people, prove using the pigeonhole principle that there must be at least 51 people who celebrate their birthdays in the same month.

Accepted Answer

There must be at least 51 people who celebrate their birthdays in the same month.

Answer

The pigeonhole principle cannot be applied here because there are 12 months and only 100 people.

Answer

There must be at least 10 people who share the same birth month.

Answer

There must be at least 51 people who share the same birthdate.

Question 2

In a group of 23 people, use the pigeonhole principle to demonstrate that at least two of them must share the same birth month.

Accepted Answer

Since there are 12 months and 23 people, the pigeonhole principle implies that at least two people must have the same birth month.

Answer

The pigeonhole principle can be used to show that there must be at least two people born in the same week.

Answer

The pigeonhole principle does not apply to this scenario because there are more people than months.

Answer

If there were exactly 12 people in the group, the pigeonhole principle would guarantee that at least two people were born in the same month.

Question 3

Which of the following is NOT a property of a graph?

Accepted Answer

Edges connect vertices in both directions.

Answer

It has a finite set of vertices and edges.

Answer

It can contain cycles.

Answer

It can model relationships between objects.

Question 4

What is the number of edges in a complete graph with 'n' vertices?

Accepted Answer

n(n-1)/2

Answer

n^2

Answer

2n

Answer

n(n+1)

Question 5

What is the chromatic number of a graph that can be coloured with 3 colours?

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 6

Which of the following is a prime number?

Accepted Answer

11

Answer

1

Answer

10

Answer

12

Question 7

What is the order of the group (ℤ₆, +)?

Accepted Answer

6

Answer

2

Answer

3

Answer

4

Question 8

In a set of 9 distinct objects, how many unique combinations of 3 objects can be formed?

Accepted Answer

252

Answer

12

Answer

84

Answer

504

Question 9

Which of the following numbers **is not** prime?

Accepted Answer

15

Answer

11

Answer

19

Answer

23

Question 10

From a set of 5 distinct objects, how many ways can you select 3 objects?

Accepted Answer

10

Answer

5

Answer

15

Answer

20

Question 11

Determine the greatest common divisor (GCD) of 12 and 18.

Accepted Answer

3

Answer

2

Answer

6

Answer

12

Question 12

Identify the definition of a Hamiltonian cycle.

Accepted Answer

A cycle that traverses every vertex in a graph exactly once

Answer

A path that traverses every vertex in a graph exactly once

Answer

A cycle that traverses every edge in a graph exactly once

Answer

A path that traverses every edge in a graph exactly once

Question 13

Define a bipartite graph.

Accepted Answer

A graph where vertices can be divided into two sets, with edges connecting only vertices from different sets

Answer

A graph where every vertex has the same degree

Answer

A graph where every vertex is adjacent to every other vertex

Answer

A graph with no cycles

Question 14

Identify the prime number from the given options.

Accepted Answer

13

Answer

12

Answer

15

Answer

17

Question 15

Which of the following is NOT a fundamental component of a graph?

Accepted Answer

A set of colors

Answer

A set of vertices

Answer

A set of edges

Answer

A set of labels

Question 16

In a combination, is the order of elements significant?

Accepted Answer

No

Answer

Yes

Question 17

The number of ways to select 5 objects from a set of 10 objects is calculated using which formula?

Accepted Answer

10C5

Answer

5C5

Answer

5! + 10!

Answer

10! - 5!

Question 18

For a relation to be reflexive, which condition must be met?

Accepted Answer

Each element is related to itself.

Answer

Each element is related to every other element.

Answer

The relation is symmetric.

Answer

The relation is transitive.

Question 19

What is the order of the group Z₆?

Accepted Answer

6

Answer

2

Answer

4

Answer

12

Question 20

Which of the following conditions is essential for a lattice to be distributive?

Accepted Answer

For all a, b, c ∈ L, a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)

Answer

For all a, b, c ∈ L, a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)

Answer

The lattice must be complete.

Answer

The lattice must be bounded.

Question 21

What is the number of edges in a complete graph with n vertices?

Accepted Answer

n(n-1)/2

Answer

n

Answer

n(n+1)

Answer

n^2

Question 22

Which of the following is a prime number?

Accepted Answer

17

Answer

6

Answer

12

Answer

25

Question 23

What is the chromatic number of a graph?

Accepted Answer

The minimum number of colors needed to color the vertices such that adjacent vertices have different colors.

Answer

The maximum number of colors needed to color the edges such that adjacent edges have different colors.

Answer

The number of vertices in the graph.

Answer

The number of edges in the graph.

Question 24

What is the principle of mathematical induction used for?

Accepted Answer

To prove that a statement is true for all natural numbers.

Answer

To prove that a statement is true for all integers.

Answer

To prove that a statement is true for all rational numbers.

Answer

To prove that a statement is true for all real numbers.

Question 25

What is the Euler characteristic of a planar graph with v vertices, e edges, and f faces?

Accepted Answer

v - e + f

Answer

v + e + f

Answer

v - e - f

Answer

v + e - f

Question 26

What does Fermat's Little Theorem state?

Accepted Answer

For any prime number p, and any integer a not divisible by p, a^(p-1) - 1 is divisible by p.

Answer

For any integer a, a^p - a is divisible by p, where p is a prime number.

Answer

For any prime number p, and any integer a, a^p - a is divisible by p.

Answer

Fermat's Little Theorem does not exist.

Question 27

Which of the following is NOT a fundamental concept in discrete mathematics?

Accepted Answer

Calculus

Answer

Graph theory

Answer

Combinatorics

Answer

Number theory

Question 28

Which of the following is a composite number?

Accepted Answer

15

Answer

7

Answer

11

Answer

19

Question 29

The formula for the sum of the first n positive integers is:

Accepted Answer

n(n+1)/2

Answer

n(n-1)/2

Answer

(n+1)(n-1)/2

Answer

n^2/2

Question 30

Which of the following is an example of a lattice?

Accepted Answer

The set of all subsets of a given set, ordered by inclusion

Answer

The set of all real numbers, ordered by magnitude

Answer

The set of all complex numbers, ordered by absolute value

Answer

The set of all prime numbers, ordered by divisibility

Question 31

Which of the following is a valid topological ordering of the directed acyclic graph (A, B, C, D, E) with edges (A, B), (B, C), (C, D), (D, E)?

Accepted Answer

A -> B -> C -> D -> E

Answer

B -> A -> C -> D -> E

Answer

D -> C -> B -> A -> E

Answer

E -> D -> C -> B -> A

Question 32

Which of the following proof techniques involves assuming the negation of a statement and deriving a contradiction?

Accepted Answer

Proof by contradiction

Answer

Mathematical induction

Answer

Proof by exhaustion

Answer

Counterexample

Question 33

The independence number of a graph is the maximum number of:

Accepted Answer

Vertices that are not connected by an edge

Answer

Edges that are not connected to any vertex

Answer

Faces that are not adjacent to any other face

Answer

Paths that do not intersect any other path

Question 34

In graph theory, what is the degree of a vertex?

Accepted Answer

The number of edges that meet at that vertex.

Answer

The number of vertices connected to that vertex.

Answer

The number of edges in the graph.

Answer

The number of vertices in the graph.

Question 35

Which statement accurately describes Hamiltonian cycles?

Accepted Answer

A Hamiltonian cycle traverses each vertex in a graph precisely once.

Answer

Every graph possesses a Hamiltonian cycle.

Answer

Finding Hamiltonian cycles in graphs is computationally efficient.

Answer

Hamiltonian cycles are exclusive to complete graphs.

Question 36

In discrete mathematics, a graph consists of:

Accepted Answer

A set of vertices and edges connecting them

Answer

A sequence of numbers following specific rules

Answer

A collection of elements related by a binary operation

Answer

A function mapping inputs to outputs

Question 37

For a graph to be connected, which condition must hold?

Accepted Answer

There's a path connecting every pair of vertices

Answer

Every vertex has a degree of at least 1

Answer

The graph is acyclic (contains no cycles)

Answer

The graph is complete (all vertices are connected)

Question 38

What is the count of spanning trees in a complete graph with 'n' vertices?

Accepted Answer

n^(n-2)

Answer

n

Answer

n^2

Answer

2^n

Question 39

How many ways can you select 5 objects from a set of 10?

Accepted Answer

252

Answer

5

Answer

10

Answer

1024

Question 40

Which number among the following is prime?

Accepted Answer

23

Answer

12

Answer

36

Answer

49

Question 41

What is the greatest common divisor (GCD) of 12 and 18?

Accepted Answer

3

Answer

2

Answer

6

Answer

9

Question 42

Find the least common multiple (LCM) of 4 and 6.

Accepted Answer

12

Answer

8

Answer

24

Answer

48

Question 43

What is the chromatic number of a graph?

Accepted Answer

The minimum number of colors required to color the vertices so that no two adjacent vertices have the same color

Answer

The maximum number of colors required to color the edges so that no two adjacent edges have the same color

Answer

The number of vertices in the graph

Answer

The number of edges in the graph

Question 44

Which of the following is a Hamiltonian path?

Accepted Answer

A path that visits every vertex in the graph exactly once

Answer

A path that starts and ends at the same vertex

Answer

A path that is as short as possible

Answer

A path that has the maximum number of edges

Question 45

What is the principle of inclusion-exclusion?

Accepted Answer

A method for counting the number of elements in a set

Answer

A method for finding the probability of an event

Answer

A method for solving linear equations

Answer

A method for proving mathematical theorems

Question 46

What is the binomial coefficient?

Accepted Answer

The number of ways to choose r elements from a set of n elements

Answer

The number of permutations of n elements

Answer

The number of combinations of n elements

Answer

The number of subsets of n elements

Question 47

Which of the following is a prime number?

Accepted Answer

11

Answer

1

Answer

6

Answer

100

Question 48

What is the greatest common divisor (GCD) of 12 and 18?

Accepted Answer

3

Answer

2

Answer

6

Answer

12

Question 49

What is the inverse of 5 modulo 11?

Accepted Answer

3

Answer

2

Answer

5

Answer

7

Question 50

How many different ways can you arrange 5 distinct objects in a row?

Accepted Answer

5!

Answer

5

Answer

10

Answer

120

Question 51

Which of the following is NOT a fundamental concept of discrete mathematics?

Accepted Answer

Calculus

Answer

Graph theory

Answer

Combinatorics

Answer

Number theory

Question 52

A graph lacking cycles is classified as a(n):

Accepted Answer

Acyclic graph

Answer

Cyclic graph

Answer

Complete graph

Answer

Planar graph

Question 53

In a permutation of 5 objects, how many arrangements are possible?

Accepted Answer

120

Answer

5

Answer

25

Answer

600

Question 54

Modular arithmetic is characterized by:

Accepted Answer

Numbers 'wrapping around'

Answer

Multiplication by a constant

Answer

Addition by a constant

Answer

Division by a constant

Question 55

Which condition guarantees a connected graph?

Accepted Answer

A path exists between every pair of vertices

Answer

Every vertex has a degree of at least 1

Answer

The graph contains no cycles

Answer

The graph is planar

Question 56

Which of the following is NOT a property of a graph?

Accepted Answer

It can have an infinite number of vertices.

Answer

It consists of a set of vertices and edges.

Answer

It can have multiple edges between the same two vertices.

Answer

It can have directed edges.

Question 57

What is the principle of inclusion-exclusion?

Accepted Answer

A method to count the number of elements in a set by summing the number of elements in subsets and subtracting the number of elements in their intersections

Answer

A method to calculate the probability of an event by adding the probabilities of individual outcomes

Answer

A method to compute the expected value of a random variable by adding the expected values of its components

Answer

A method to calculate the variance of a random variable by subtracting the expected value from its components

Question 58

Which of the following is a necessary condition for a set of vectors to be linearly independent?

Accepted Answer

The vectors must not be all multiples of a single vector.

Answer

The vectors must be all zero vectors.

Answer

The vectors must be all orthogonal to each other.

Answer

The number of vectors must be less than the dimension of the vector space.

Question 59

Which of the following is NOT a valid type of graph in graph theory?

Accepted Answer

Directed Hypergraph

Answer

Undirected Graph

Answer

Directed Graph

Answer

Weighted Graph

Question 60

In a complete graph with n vertices, how many edges are present?

Accepted Answer

n(n-1)/2

Answer

n^2

Answer

n(n+1)/2

Answer

n^3

Question 61

Which statement about prime numbers is accurate?

Accepted Answer

A prime number is divisible by 1 and itself only.

Answer

A prime number must be less than 10.

Answer

A prime number is always an even number.

Answer

A prime number is always a multiple of 5.

Question 62

What is the chromatic number of any bipartite graph?

Accepted Answer

2

Answer

1

Answer

3

Question 63

In the context of mathematical induction, which of the following is a valid induction step?

Accepted Answer

Assuming the statement holds true for n, prove that it also holds true for n+1.

Answer

Assuming the statement holds true for n, prove that it also holds true for n-1.

Answer

Assuming the statement holds true for n, prove that it also holds true for 2n.

Question 64

What is the order of the group Z/12Z?

Accepted Answer

12

Answer

4

Answer

6

Answer

2