Discrete Mathematics

Discrete Mathematics is the branch of mathematics concerned with countable, distinct, and separate objects, forming the foundational language of computer science and information technology. Unlike continuous mathematics, which deals with smoothly varying quantities like those in calculus, discrete math explores fundamental concepts such as logic, set theory, combinatorics (the art of counting), and graph theory. These principles are essential for modeling, designing, and analyzing computer algorithms, data structures, relational databases, and communication networks, providing the tools to solve problems in a finite, step-by-step manner.

  1. Foundations of Logic and Proofs
    1. Propositional Logic
      1. Basic Concepts
        1. Definition of a Proposition
          1. Atomic Propositions
            1. Compound Propositions
              1. Statement vs Non-Statement
              2. Logical Connectives
                1. Negation
                  1. Conjunction
                    1. Disjunction
                      1. Exclusive Or
                        1. Implication
                          1. Biconditional
                          2. Syntax and Semantics
                            1. Well-Formed Formulas
                              1. Precedence of Operators
                                1. Parentheses and Grouping
                                2. Truth Tables
                                  1. Constructing Basic Truth Tables
                                    1. Truth Tables for Complex Formulas
                                      1. Analyzing Logical Statements
                                      2. Logical Equivalence
                                        1. Definition and Testing
                                          1. Fundamental Equivalences
                                            1. De Morgan's Laws
                                              1. Distributive Laws
                                                1. Absorption Laws
                                                  1. Double Negation
                                                    1. Tautologies
                                                      1. Contradictions
                                                        1. Contingencies
                                                        2. Conditional Statements
                                                          1. Structure of Conditionals
                                                            1. Converse
                                                              1. Contrapositive
                                                                1. Inverse
                                                                  1. Necessary and Sufficient Conditions
                                                                  2. Argument Forms
                                                                    1. Valid Arguments
                                                                      1. Invalid Arguments
                                                                        1. Sound Arguments
                                                                        2. Translation from Natural Language
                                                                          1. Identifying Propositions in English
                                                                            1. Common Translation Patterns
                                                                              1. Ambiguity Resolution
                                                                              2. Satisfiability and Validity
                                                                                1. Satisfiable Formulas
                                                                                  1. Valid Formulas
                                                                                    1. Unsatisfiable Formulas
                                                                                  2. Predicate Logic
                                                                                    1. Predicates and Variables
                                                                                      1. Definition of Predicates
                                                                                        1. Propositional Functions
                                                                                          1. Domain of Variables
                                                                                          2. Quantifiers
                                                                                            1. Universal Quantifier
                                                                                              1. Existential Quantifier
                                                                                                1. Restricted Quantifiers
                                                                                                2. Quantified Statements
                                                                                                  1. Simple Quantified Statements
                                                                                                    1. Bound Variables
                                                                                                      1. Free Variables
                                                                                                        1. Scope of Quantifiers
                                                                                                        2. Multiple Quantifiers
                                                                                                          1. Order of Quantifiers
                                                                                                            1. Nested Quantifiers
                                                                                                              1. Mixed Quantifiers
                                                                                                              2. Negation of Quantified Statements
                                                                                                                1. Negating Universal Statements
                                                                                                                  1. Negating Existential Statements
                                                                                                                    1. De Morgan's Laws for Quantifiers
                                                                                                                    2. Translation to Predicate Logic
                                                                                                                      1. English to Predicate Logic
                                                                                                                        1. Mathematical Statements
                                                                                                                          1. Complex Statements
                                                                                                                          2. Equivalences in Predicate Logic
                                                                                                                            1. Quantifier Equivalences
                                                                                                                              1. Prenex Normal Form
                                                                                                                            2. Rules of Inference
                                                                                                                              1. Propositional Inference Rules
                                                                                                                                1. Modus Ponens
                                                                                                                                  1. Modus Tollens
                                                                                                                                    1. Hypothetical Syllogism
                                                                                                                                      1. Disjunctive Syllogism
                                                                                                                                        1. Addition
                                                                                                                                          1. Simplification
                                                                                                                                            1. Conjunction
                                                                                                                                              1. Resolution
                                                                                                                                              2. Quantifier Inference Rules
                                                                                                                                                1. Universal Instantiation
                                                                                                                                                  1. Universal Generalization
                                                                                                                                                    1. Existential Instantiation
                                                                                                                                                      1. Existential Generalization
                                                                                                                                                      2. Formal Proof Construction
                                                                                                                                                        1. Proof Structure
                                                                                                                                                          1. Justification of Steps
                                                                                                                                                            1. Direct Application of Rules
                                                                                                                                                            2. Common Logical Fallacies
                                                                                                                                                              1. Affirming the Consequent
                                                                                                                                                                1. Denying the Antecedent
                                                                                                                                                                  1. Hasty Generalization
                                                                                                                                                                    1. Existential Fallacy
                                                                                                                                                                  2. Proof Techniques
                                                                                                                                                                    1. Proof Fundamentals
                                                                                                                                                                      1. Axioms
                                                                                                                                                                        1. Definitions
                                                                                                                                                                          1. Theorems
                                                                                                                                                                            1. Lemmas
                                                                                                                                                                              1. Corollaries
                                                                                                                                                                                1. Conjectures
                                                                                                                                                                                2. Direct Proof
                                                                                                                                                                                  1. Structure and Method
                                                                                                                                                                                    1. Examples in Number Theory
                                                                                                                                                                                      1. Examples in Set Theory
                                                                                                                                                                                      2. Proof by Contraposition
                                                                                                                                                                                        1. Method and Structure
                                                                                                                                                                                          1. When to Use Contraposition
                                                                                                                                                                                            1. Examples and Applications
                                                                                                                                                                                            2. Proof by Contradiction
                                                                                                                                                                                              1. Method and Structure
                                                                                                                                                                                                1. Reductio ad Absurdum
                                                                                                                                                                                                  1. Examples and Applications
                                                                                                                                                                                                  2. Proof by Cases
                                                                                                                                                                                                    1. Exhaustive Case Analysis
                                                                                                                                                                                                      1. Constructing Cases
                                                                                                                                                                                                        1. Examples and Applications
                                                                                                                                                                                                        2. Trivial and Vacuous Proofs
                                                                                                                                                                                                          1. Trivial Proofs
                                                                                                                                                                                                            1. Vacuous Proofs
                                                                                                                                                                                                              1. Recognition and Application
                                                                                                                                                                                                              2. Existence Proofs
                                                                                                                                                                                                                1. Constructive Existence Proofs
                                                                                                                                                                                                                  1. Non-constructive Existence Proofs
                                                                                                                                                                                                                  2. Uniqueness Proofs
                                                                                                                                                                                                                    1. Direct Uniqueness Proofs
                                                                                                                                                                                                                      1. Proof by Contradiction for Uniqueness
                                                                                                                                                                                                                      2. Biconditional Proofs
                                                                                                                                                                                                                        1. Proving Both Directions
                                                                                                                                                                                                                          1. Chain of Equivalences
                                                                                                                                                                                                                          2. Counterexamples
                                                                                                                                                                                                                            1. Finding Counterexamples
                                                                                                                                                                                                                              1. Disproving Universal Statements
                                                                                                                                                                                                                              2. Mathematical Induction
                                                                                                                                                                                                                                1. Principle of Mathematical Induction
                                                                                                                                                                                                                                  1. Base Case
                                                                                                                                                                                                                                    1. Inductive Step
                                                                                                                                                                                                                                      1. Inductive Hypothesis
                                                                                                                                                                                                                                        1. Strong Induction
                                                                                                                                                                                                                                          1. Well-Ordering Principle
                                                                                                                                                                                                                                            1. Common Induction Mistakes

                                                                                                                                                                                                                                        Go to top

                                                                                                                                                                                                                                        Next

                                                                                                                                                                                                                                        2. Set Theory and Basic Structures