math-topics

• Mathematical Logic

  • Logical System

    • Logical Argument

      • Proposition, Atom, Formula, Hypothesis

      • Colorally, Axiom, Theorem, Lemma

      • Declarative statement

        • Premise, Conclusion

      • Conditional statement

        • Antecedent, Consequent

        • Sufficiency, Necessity

      • Deductive Argument (necessity)

        • Deductive argument forms

          • argument based on mathematics

          • argument from definition

          • categorical syllogism

          • hypothetical syllogism

            • pure hypothetical syllogism

            • mixed hypothetical syllogism

          • disjunctive syllogism

      • Inductive Argument (probability)

        • Inductive Argument Forms

          • prediction

          • argument from analogy

          • generalization

          • argument from authority

          • argument based on signs

          • causal inference

      • Logical Form

        • Logical Truth

        • Validity

        • Soundness

        • Strength

        • Cogency

    • Logic Interpretation

    • Logical Laws

      • Law of Identity

      • Law of Excluded Middle

      • Tautology

      • Fallacy

      • Contradictions

      • Contingency

      • Propositional Equivalences

      • Inverse, Converse, and Contra-positive

      • Duality Principle

      • Operators and Postulates

        • Closure

        • Associative Laws

        • Commutative Laws

        • Distributive Laws

        • Identity

        • Inverse

        • De Morgan's Law

    • Logical Connectives

      • negation

      • conjunction

      • disjunction

      • implication

      • bijunction

      • Normal Forms: CNF, DNF

    • Inference rules

      • conjunction elimination

      • conjunction introduction (1,2)

      • disjunction elimination, introduction

      • negation elimination, introduction, double negation

      • implication elimination, introduction

      • Modus Ponens, Modus Tollens

      • Addition

      • Conjunction

      • Simplification

      • MP, MT

      • Disjunctive Syllogism

      • Hypothetical Syllogism

      • Constructive Dilemma

      • Destructive Dilemma

    • Sequent Calculus

      • antecedent

      • consequent

    • Structural Rules

      • Weakening

      • Exchange

      • Contraction

  • Logic Systems

    • Formal Logic Systems

      • Symbol

      • Consistency

      • Validity

      • Soundness

      • Completeness

      • Well-Orderedness

    • Syllogistic Logic

    • Propositional Logic

      • Well Formed Formula

      • Quantifiers

    • First-order Logic

      • First-Order Theories

        • Formal systems

        • First-order languages and theories

        • Tautologies

        • Theorems and rules in first-order theories

      • Consistency

        • Consistency and completeness

        • Extensions by definitions

        • Interpretations

        • Herbrand–Skolem theory

        • Craig’s interpolation lemma

      • Incompleteness theorem

        • Number-theoretic functions and predicates

        • Representability

        • Arithmetizations

        • The incompleteness theorem

        • Minimal arithmetic

      • First-Order Number theory

        • Recursive extensions

        • The first-order theory PA

    • Second-order Logic

    • Higher-order Logic

    • Multi-valued Logic

    • Substructural Logic

      • Relevant Logic

      • Linear Logic

    • Separation Logic

    • Temporal Logic

    • Model Logic

• Counting Theory

  • The Rules of Sum and Product

  • Permutations

  • Combinations

  • Pascal's Identity

  • Pigeonhole Principle

  • The Inclusion-Exclusion principle

  • Probability

    • Probability Axioms

    • Properties of Probability

    • Conditional Probability

    • Bayes' Theorem

• Algebraic Structures

  • Mathematical induction

    • Induction

    • Strong Induction

  • Recurrence relations

    • Linear Recurrence Relations

    • Non-Homogeneous Recurrence Relation and Particular Solutions

    • Generating Functions

  • Graph theory

    • Graph

    • Graph Models

    • Types of Graphs

    • Representation of Graphs

    • Planar vs. Non-planar graph

    • Isomorphism

    • Homomorphism

    • Euler Graphs

    • Hamiltonian Graphs

    • Graph Coloring

    • Traversal

  • Trees

    • Properties

    • Centers and Bi-Centers of a Tree

    • Labeled Trees

    • Unlabeled Trees

    • Rooted Tree

    • Binary Search Tree

    • Spanning Trees

    • Kruskal's Algorithm

    • Prim's Algorithm

  • Group theory

    • Semigroup

    • Monoid

    • Group

    • Abelian Group

    • Cyclic Group and Subgroup

    • Partially Ordered Set (POSET)

    • Linearly Ordered Set

    • Hasse Diagram

    • Lattice

  • Field

  • Ring

  • Boolean algebra

    • Boolean Functions

    • Boolean Expressions

    • Boolean Identities

    • Canonical Forms

    • Logic Gates

    • Simplification Of Boolean Functions

      • Simplification Using Algebraic Functions

      • Karnaugh Maps

      • Simplification Using K-map

• Theory of Computation

  • Logic in CS

    • Logic and verification

    • Proof theory

    • Modal and temporal logics

    • Automated reasoning

    • Constraint and logic programming

    • Constructive mathematics

    • Description logics

    • Equational logic and rewriting

    • Finite Model Theory

    • Higher order logic

    • Linear logic

    • Programming logic

    • Abstraction

    • Verification by model checking

    • Type theory

    • Hoare logic

    • Separation logic

  • Theory and algorithms for application domains

    • Program semantics

      • Algebraic semantics

      • Denotational semantics

      • Operational semantics

      • Axiomatic semantics

      • Action semantics

      • Categorical semantics

    • Program reasoning

      • Invariants

      • Program specifications

      • Pre- and post-conditions

      • Program verification

      • Program analysis

      • Assertions

      • Parsing

      • Abstraction

  • Lambda Calculi

    • Lambda Calculus

    • Simply Typed Lambda Calculus

    • System F

    • Lambda Pi

    • Lambda Omega

    • Lambda Omega Pi

  • Formal Language

    • Syntax

    • Semantics

    • Grammer

    • BNF

  • Complexity Theory

  • Automata Theory

    • DFA

    • NFA

  • Halting Problem