Discrete Mathematics

IGCSE

IB

USA

Singapore

Primary Year

Secondary Year

Junior College

Poly Technique

College

Course Content

UNIT 1: The Foundations: Logic and Proofs.

1.1 Propositional Logic
1.2 Applications of Propositional Logic
1.3 Propositional Equivalences
1.4 Predicates and Quantifiers
1.5 Nested Quantifiers
1.6 Rules of Inference
1.7 Introduction to Proofs
1.8 Proof Methods and Strategy

UNIT 2: Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

UNIT 3: Relations

UNIT 4: Number Theory

UNIT 5: Induction and Recursion

UNIT 6: Counting

UNIT 7: Advanced Counting Techniques

UNIT 8: Discrete Probability

UNIT 9: Algorithms

UNIT 10: Graphs

UNIT 11: Trees

11.5 Minimum Spanning Trees

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IGCSE

Primary Year

Secondary Year

Junior College

Poly Technique

Course Content

UNIT 1: The Foundations: Logic and Proofs.

1.1 Propositional Logic

1.2 Applications of Propositional Logic

1.3 Propositional Equivalences

1.4 Predicates and Quantifiers

1.5 Nested Quantifiers

1.6 Rules of Inference

1.7 Introduction to Proofs

1.8 Proof Methods and Strategy

UNIT 2: Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

2.1 Sets

2.2 Set Operations.

2.3 Functions

2.4 Sequences and Summations

2.5 Cardinality of Sets.

2.6 Matrices

UNIT 3: Relations

3.1 Relations and Their Properties

3.2 n-ary Relations and Their Applications Representing Relations

3.3 Closures of Relations

3.4 Equivalence Relations.

3.5 Partial Orderings

UNIT 4: Number Theory

4.1 Divisibility and Modular Arithmetic

4.2 Integer Representations and Algorithms

4.3 Primes and Greatest Common Divisors

4.4 Solving Congruences

4.5 Applications of Congruences

UNIT 5: Induction and Recursion

5.1 Mathematical Induction

5.2 Strong Induction and Well-Ordering

5.3 Recursive Definitions and Structural Induction

5.4 Recursive Algorithms

5.5 Program Correctness

UNIT 6: Counting

6.1 The Basics of Counting.

6.2 The Pigeonhole Principle

6.3 Permutations and Combinations…

6.4 Binomial Coefficients and Identities

6.5 Generalized Permutations and Combinations

6.6 Generating Permutations and Combinations

UNIT 7: Advanced Counting Techniques

7.1 Applications of Recurrence Relations

7.2 Solving Linear Recurrence Relations

7.3 Divide-and-Conquer Algorithms and Recurrence Relations

7.4 Generating Functions

7.5 Inclusion-Exclusion

7.6 Applications of Inclusion-Exclusion

UNIT 8: Discrete Probability

8.1 An Introduction to Discrete Probability

8.2 Probability Theory.

8.3 Bayes’ Theorem

8.4 Expected Value and Variance

UNIT 9: Algorithms

9.1: Big-O Notation

9.2: Some Algorithms

9.3 The growth of functions

UNIT 10: Graphs

10.1 Graphs and Graph Models

10.2 Graph Terminology and Special Types of Graphs

10.3 Representing Graphs and Graph Isomorphism

10.4 Connectivity

10.5 Euler and Hamilton Paths

10.6 Shortest-Path Problems

10.7 Planar Graphs

10.8 Graph Coloring.

UNIT 11: Trees

11.1 Introduction to Trees

11.2 Applications of Trees

11.3 Tree Traversal

11.4 Spanning Trees.

11.5 Minimum Spanning Trees

11.5 Minimum Spanning Trees