Topos Math Library

Topos is a library for implementations of mathematical concepts for .NET Standard 2.0 environment. Based on Zermelo–Fraenkel set theory (ZFC). Currently only supports finite sets.

A Set is an unordered container of mathematical objects, including nested definitions such as Set of Sets. My implementation takes .NET HashSet<T> as basis. However, Sets are not generic types, and can only hold objects of MathObject class.

ZFC ensures that there are no atomic elements, however, to increase comprehension, I included atomic elements where Element is its base class.

Currently supported classes are:

Topos.Core

• MathObject (abstract),
  • Element
    • Indeterminate
    • Number (abstract)
      • Real
        • Integer
          • Natural
        • Rational
      • Complex
    • Exponential
  • Set
    • GeneratedSet
    • OrderedTuple
    • BinaryRelation
      • Function

Topos.Core.Generic

• MathObject (from Topos.Core)
  • GenericSet<T>

Topos.Core.Exceptions

• Exception (.NET)
  • ToposException
    • ArgumentCountException
    • DimensionMismatchException
    • IndeterminateException
    • UndefinedDomainException
      • ComplexDomainException

Topos.NumberTheory

• ICongruence<T> (interface)
• MathObject (from Topos.Core)
  • IntegerCongruence
• Division (static)
• Fibonacci (static)
• NumberTheoreticFunctions (static)
• Primality (static)

TO-DO:

Topos.Core:

• Exponentials will be represented as numbers, including complex number operations (will not support Indeterminates)
• Infinite sets (Countably - Uncountably)

Topos.NumberTheory:

• Modular arithmetic over integers
  • Finding solutions of x^2 ≡ a (mod n)
• Linear Diophantine equations
• Aliquot sums, perfect numbers and other related concepts
• Sums of squares
• Continued fractions

ISSUES:

• Complex number operations between ordered tuples are not supported.
• Complex number operations over exponential representations are not supported.