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Quizzes > Mathematics & Statistics

Basic Discrete Mathematics Quiz

Free Practice Quiz & Exam Preparation

Difficulty: Moderate
Questions: 15
Study OutcomesAdditional Reading
3D voxel art representing Basic Discrete Mathematics course content

Boost your learning with our Basic Discrete Mathematics practice quiz, designed to reinforce essential concepts like sets and relations, functions, and basic counting techniques. Dive into challenging questions on recurrence relations, graphs and trees, and matrix algebra to sharpen your problem-solving skills and enhance your understanding of algorithm efficiency in discrete mathematics.

Which of the following best defines a set?
A matrix of numbers
A collection of distinct objects
A group of numbers that are consecutive
A system of equations
A set is defined as a well-defined collection of distinct objects where order does not matter and duplicates are not allowed. This distinguishes sets from other mathematical structures like sequences or matrices.
Which of the following best describes a function?
A relation where each element of the domain is associated with exactly one element of the codomain
An unordered collection of elements
A process that randomly pairs elements from two sets
A mapping that assigns multiple values to a single domain element
A function is defined as a relation in which every input has a unique output. Option A captures the essential property of functions, ensuring a one-to-one association from domain to codomain.
In counting problems, which technique is appropriate when order is not important?
Permutations
Recurrence relations
Cartesian products
Combinations
Combinations are used when the order of selection does not matter, which makes them appropriate for such counting problems. In contrast, permutations are used when order is significant.
What is the primary characteristic of a tree in graph theory?
It always contains cycles
It is a complete graph
It is an acyclic connected graph
It can have isolated vertices
A tree is a connected graph that does not contain any cycles, ensuring there is exactly one path between any two vertices. This acyclic nature distinguishes trees from other types of graphs.
Which of the following properties does matrix algebra generally not have?
Commutativity of matrix multiplication
Existence of inverse for non-singular matrices
Associativity of matrix multiplication
Distributivity over matrix addition
Matrix multiplication is generally not commutative, meaning the order of multiplication matters. The other properties like associativity and distributivity are standard in matrix algebra when the matrices meet the necessary conditions.
Consider the recurrence relation T(n) = T(n-1) + 2 with T(1) = 3. What is the value of T(5)?
10
11
13
12
By applying the recurrence relation step by step: T(2)=5, T(3)=7, T(4)=9, and finally T(5)=11. The linear addition of 2 at each step leads directly to the answer.
Which of the following defines an equivalence relation on a set?
A relation that is reflexive and antisymmetric
A relation that is reflexive, symmetric, and transitive
A relation that is only symmetric
A relation that is only transitive
An equivalence relation must satisfy three properties: reflexivity, symmetry, and transitivity. Option A correctly lists all necessary conditions for an equivalence relation.
Given the functions f(x) = 2x + 1 and g(x) = x², what is the result of the composition (f ∘ g)(x)?
2x + 1
2x² + 1
4x² + 1
x² + 2
The composition (f ∘ g)(x) means applying g(x) first and then f(x). Substituting x² into f gives f(x²) = 2x² + 1, which is the correct result.
Which principle is typically used to compute the total number of outcomes when making a series of independent choices?
Fundamental Counting Principle
Inclusion-Exclusion Principle
Pigeonhole Principle
Principle of Mathematical Induction
The Fundamental Counting Principle states that if there are multiple independent choices, the total number of outcomes is the product of the number of possibilities for each choice. This principle is the basis for many counting problems in discrete mathematics.
Which algorithm is most appropriate for finding the shortest path in a weighted graph with non-negative edge weights?
Kruskal's algorithm
Bellman-Ford algorithm
Depth-first search
Dijkstra's algorithm
Dijkstra's algorithm is designed to find the shortest paths in graphs with non-negative edge weights efficiently. While the Bellman-Ford algorithm can handle negative weights, it is less efficient when only non-negative weights are present.
What is the determinant of a 2×2 matrix [[a, b], [c, d]]?
ac - bd
ab - cd
bc - ad
ad - bc
The determinant of a 2×2 matrix is calculated using the formula ad - bc. This fundamental operation in matrix algebra determines properties such as invertibility and is widely used in linear transformations.
Which proof technique involves verifying a base case and then proving that if a statement holds for an integer k, it holds for k+1?
Proof by contradiction
Direct proof
Mathematical induction
Proof by exhaustion
Mathematical induction is a method of proof consisting of two steps: validating the base case and proving that if the statement holds for an arbitrary integer k, it holds for k+1. This technique is particularly effective for proving propositions pertaining to natural numbers.
For a graph represented by an adjacency matrix, which property indicates that the graph is undirected?
The matrix is symmetric
The matrix is invertible
The matrix has zeros on the main diagonal
The matrix is diagonal
An undirected graph is characterized by an adjacency matrix that is symmetric, meaning that the entry for row i, column j is equal to the entry for row j, column i. This symmetry reflects the bidirectional nature of the connections in an undirected graph.
What is the relationship between the trace of a square matrix and its eigenvalues?
The trace equals the maximum eigenvalue
The trace is equal to the sum of its eigenvalues
The trace equals the minimum eigenvalue
The trace is equal to the product of its eigenvalues
The trace of a square matrix, defined as the sum of its diagonal elements, is also equal to the sum of its eigenvalues (counting multiplicities). This important relation connects two fundamental concepts in matrix theory.
Which of the following is an example of an antisymmetric relation on a set?
The subset relation (⊆) on sets
The relation 'has the same birthday as'
The isomorphism relation on graphs
The parallelism relation between lines in a plane
The subset relation (⊆) on sets is antisymmetric because if one set is a subset of another and vice versa, the two sets must be equal. The other listed relations either do not satisfy antisymmetry or are inherently symmetric.
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Study Outcomes

  1. Analyze sets and relations to determine key properties and operations.
  2. Evaluate functions by examining injectivity, surjectivity, and composition.
  3. Apply counting techniques and recurrence relations to solve combinatorial problems.
  4. Interpret graphs and tree structures to assess connectivity and algorithm efficiency.
  5. Utilize matrix algebra to model and solve discrete mathematical problems.

Basic Discrete Mathematics Additional Reading

Here are some top-notch resources to supercharge your discrete mathematics journey:

  1. Mathematics for Computer Science Dive into MIT's comprehensive course covering sets, relations, functions, counting techniques, recurrence relations, graphs, trees, and matrix algebra. It's a treasure trove of knowledge!
  2. Discrete Mathematics Join Shanghai Jiao Tong University's Coursera course, offering 11 modules on discrete math fundamentals, including algorithms and their efficacy. Perfect for flexible, self-paced learning.
  3. Combinatorial Theory: Introduction to Graph Theory, Extremal and Enumerative Combinatorics Explore MIT's advanced course delving into graph theory, combinatorics, and their applications. A deep dive for the mathematically adventurous.
  4. Best Online Discrete Mathematics Courses and Programs Discover a curated list of online courses from edX, covering various discrete math topics. A great way to find the perfect course tailored to your needs.
  5. Resources for Discrete Math Check out NYU's compilation of discrete math resources, including textbooks, lecture notes, and problem sets. A handy toolkit for mastering the subject.
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