How this course works
The chapters and lessons below are the fixed course structure. When you start, LearnLive teaches each lesson interactively and adapts examples, pacing, and questions to you.
Complete syllabus
Every chapter and lesson
- 01
Chapter 1 · 3 lessons
Logic & Proofs
- 1.1
Propositional logic
9 min · Quick check
Lesson goal: By the end you can understand the fundamentals of propositional logic and its components.
- Define propositional logic and its significance in mathematics.
- Identify and construct simple propositions.
- Understand logical connectives such as AND, OR, NOT, and IMPLIES.
- 1.2
Predicates & quantifiers
9 min · Quick check
Lesson goal: By the end you can explain predicates and quantifiers and their roles in logical statements.
- Define predicates and their use in logic.
- Explain the concept of quantifiers, including universal and existential quantifiers.
- Construct logical statements using predicates and quantifiers.
- 1.3
Methods of proof
9 min · Quick check
Lesson goal: By the end you can apply various methods of proof to establish the validity of mathematical statements.
- Identify different methods of proof, including direct proof, proof by contradiction, and proof by contrapositive.
- Apply these methods to solve logical problems.
- Understand the importance of proof in mathematics.
- 1.1
- 02
Chapter 2 · 3 lessons
Sets, Functions & Relations
- 2.1
Sets
8 min · Quick check
Lesson goal: By the end you can define sets and understand their properties.
- Define a set and its elements.
- Understand the notation used for sets.
- Explore different types of sets, such as finite and infinite sets.
- Learn about subsets and the power set.
- 2.2
Functions
9 min · Quick check
Lesson goal: By the end you can explain functions and their characteristics.
- Define a function and its components: domain, codomain, and range.
- Understand the concept of one-to-one, onto, and bijective functions.
- Explore function notation and how to evaluate functions.
- Learn about composite functions and inverse functions.
- 2.3
Relations
8 min · Quick check
Lesson goal: By the end you can describe relations and their properties.
- Define a relation and its representation using sets of ordered pairs.
- Understand the concepts of reflexivity, symmetry, and transitivity in relations.
- Explore equivalence relations and their properties.
- Learn about partial orders and total orders.
- 2.1
- 03
Chapter 3 · 3 lessons
Number Theory
- 3.1
Divisibility & modular arithmetic
9 min · Quick check
Lesson goal: By the end you can understand the concepts of divisibility and modular arithmetic.
- Define divisibility and its properties.
- Explain modular arithmetic and its notation.
- Illustrate the relationship between divisibility and modular arithmetic.
- 3.2
Primes & GCD
8 min · Quick check
Lesson goal: By the end you can identify prime numbers and calculate the greatest common divisor (GCD).
- Define prime numbers and their significance in number theory.
- Explain the concept of the greatest common divisor (GCD).
- Demonstrate methods to find the GCD of two numbers.
- 3.3
Applications
8 min · Quick check
Lesson goal: By the end you can apply number theory concepts to solve real-world problems.
- Identify real-world applications of divisibility and primes.
- Discuss the importance of GCD in problem-solving.
- Explore examples of modular arithmetic in computer science.
- 3.1
- 04
Chapter 4 · 3 lessons
Induction & Recursion
- 4.1
Mathematical induction
9 min · Quick check
Lesson goal: By the end you can understand and apply the principle of mathematical induction.
- Define mathematical induction and its components: base case and inductive step.
- Explain the process of proving a statement for all natural numbers using induction.
- Identify common examples of mathematical induction in proofs.
- 4.2
Recursive definitions
8 min · Quick check
Lesson goal: By the end you can formulate and analyze recursive definitions.
- Define recursive definitions and their components: base case and recursive case.
- Illustrate how recursive definitions can describe sequences and functions.
- Provide examples of recursive definitions in mathematics.
- 4.3
Recurrence relations
8 min · Quick check
Lesson goal: By the end you can solve recurrence relations and understand their applications.
- Define recurrence relations and their significance in discrete mathematics.
- Explain methods for solving linear recurrence relations.
- Discuss applications of recurrence relations in algorithm analysis.
- 4.1
- 05
Chapter 5 · 3 lessons
Counting
- 5.1
The basics of counting
8 min · Quick check
Lesson goal: By the end you can understand the fundamental principles of counting.
- Define counting and its importance in mathematics.
- Introduce the basic counting principles: addition and multiplication.
- Explain the concept of sets and elements.
- 5.2
Permutations & combinations
9 min · Quick check
Lesson goal: By the end you can differentiate between permutations and combinations and apply them in various scenarios.
- Define permutations and combinations with examples.
- Explain the formula for permutations and combinations.
- Discuss the significance of order in permutations versus combinations.
- 5.3
Pigeonhole & binomials
8 min · Quick check
Lesson goal: By the end you can apply the Pigeonhole Principle and understand binomial coefficients.
- Define the Pigeonhole Principle and provide examples.
- Introduce binomial coefficients and their notation.
- Explain the relationship between binomial coefficients and combinatorial problems.
- 5.1
- 06
Chapter 6 · 3 lessons
Graphs & Trees
- 6.1
Graphs
9 min · Quick check
Lesson goal: By the end you can define and identify different types of graphs and their components.
- Define a graph and its components: vertices and edges.
- Differentiate between directed and undirected graphs.
- Explain weighted and unweighted graphs.
- 6.2
Graph algorithms
9 min · Quick check
Lesson goal: By the end you can apply various algorithms to solve problems related to graphs.
- Introduce common graph algorithms such as Depth-First Search and Breadth-First Search.
- Explain the concept of shortest path and introduce Dijkstra's algorithm.
- Discuss the applications of graph algorithms in real-world scenarios.
- 6.3
Trees
8 min · Quick check
Lesson goal: By the end you can define trees and understand their properties and applications.
- Define a tree and its characteristics, including root, leaves, and height.
- Differentiate between binary trees and general trees.
- Explain the concept of tree traversal methods: pre-order, in-order, and post-order.
- 6.1