Topic: Set Theory

A branch of maths. The study of sets and functions between sets. A set is a collection of mathematical objects.

• inclusion in a set
• the empty set
• the singleton set

Sources

Category theory for programmers¹ touches on set theory in order to expand into category theory

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Backlinks

• Topic: Maths
  • set theory
• Video Series: Category Theory
  • set theory
• Empty set
  • In set theory, the empty set is a set with no elements.
• Codomain
  • In Maths¹, the codomain of a function is the set to which it's output belongs. The exact set of a functions output is known as its range.
• The singleton set and the unit type
  • The singleton set from set theory corresponds to the Unit type from type theory. Also to true in logic, and to a 1-category in category theory.
• Domain
  • In Maths¹, the domain of a function is the set of all possible inputs.
• Inclusion in a set
  • In set theory, a set is defined by its elements. We can state that an element is in a set using the $\\in$ notation - using \in in $\\LaTeX$.
• The empty set and the void type
  • The empty set in set theory corresponds to the Void type in type theory or to false in logic. There's also the 0-category in category theory.
• Blackboard Bold
  • In maths, a natural number is any positive integer. The set of all natural numbers is denoted $\\mathbb{N}$.
• Injectivity
  • In set theory, injectivity is a property of a function. Given a function $f : A \\rightarrow B$, if when supplied with every input in $A$ $f$ produces every possible output in $B$ then $f$ is injective.
• De-duplicate an Array by value
  • This runs it through a Set, which ensures that there's only one element of each type, and then converts it back into an array. One footgun with this approach is that Set appears to test equality by reference. While this works for primitive values, if you're trying to de-duplicate an array of complex data this isn't going to work so well.
• A pair of types with the same cardinality will always be isomorphic
  • There will always be a way of exhaustively mapping each value of type $a$ to type $b$, injectively and surjectively when viewing the domain and the codomain as a set.
• Language: Function over a set
  • In maths, for a given function, it can be said to operate 'over' a set in the case that its domain and codomain are in the same set. For example the function $f : \\mathbb{R} \\rightarrow \\mathbb{R}$ could be said to be a function over reals.
• Tuples
  • $\\mathbb{R}^3$ denotes a set of 3-tuples of real numbers, as an example. I.e $(4, 1023, -3) \\in \\mathbb{R}^3$ (inclusion in a set)
• A monoid has an associative binary operation
  • Any monoid has an associative binary operation that takes any two elements in the associated set and produces an element in that set.
• Integer
  • In maths, an integer is a whole number. The set of all integers is denoted $\\mathbb{Z}$
• Natural Number
  • In maths, a natural number is any positive integer. The set of all natural numbers is denoted $\\mathbb{N}$.
• The identity morphism in a monoidal category corresponds to the unit element
  • The unit element defined for a set-theory monoid corresponds to the identity morphism within a monoidal category.
• A set-theory monoid and a category-theory monoid are the same thing
  • The set theory definition and the category theory definition of monoids are different views over the same concept. Category theory defines a monoid as any category with a single object. Set theory defines a monoid as a set with an associative binary operation and a unit element.

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1. Category theory for programmers ↩↩↩