# 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
