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For example, if number => number is the type of function taking a number as an argument and returning a number, and string => string is the type of function taking a string as an argument and returning a string, then the intersection of these two types can be used to describe (overloaded) functions that do one or the other, based on what type ...
The intersection is the meet/infimum of and with respect to because: if L ∩ R ⊆ L {\displaystyle L\cap R\subseteq L} and L ∩ R ⊆ R , {\displaystyle L\cap R\subseteq R,} and if Z {\displaystyle Z} is a set such that Z ⊆ L {\displaystyle Z\subseteq L} and Z ⊆ R {\displaystyle Z\subseteq R} then Z ⊆ L ∩ R . {\displaystyle Z ...
So the intersection of the empty family should be the universal set (the identity element for the operation of intersection), [4] but in standard set theory, the universal set does not exist. However, when restricted to the context of subsets of a given fixed set X {\displaystyle X} , the notion of the intersection of an empty collection of ...
In this Boolean algebra, union can be expressed in terms of intersection and complementation by the formula = (), where the superscript denotes the complement in the universal set . Alternatively, intersection can be expressed in terms of union and complementation in a similar way: A ∩ B = ( A ∁ ∪ B ∁ ) ∁ {\displaystyle A\cap B ...
A Jónsson cardinal is a large cardinal such that for every function f: [κ] <ω → κ there is a set H of order type κ such that for each n, f restricted to n-element subsets of H omits at least one value in κ. 3. A Jónsson function is a function : [] with the property that, for any subset y of x with the same cardinality as x, the ...
The same fact can be stated as the indicator function (denoted here by ) of the symmetric difference, being the XOR (or addition mod 2) of the indicator functions of its two arguments: () = or using the Iverson bracket notation [] = [] [].
Languages with first-class functions have function types like "a function expecting a Cat and returning an Animal" (written cat-> animal in OCaml syntax or Func < Cat, Animal > in C# syntax). Those languages also need to specify when one function type is a subtype of another—that is, when it is safe to use a function of one type in a context ...
In computer science and logic, a dependent type is a type whose definition depends on a value. It is an overlapping feature of type theory and type systems.In intuitionistic type theory, dependent types are used to encode logic's quantifiers like "for all" and "there exists".