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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 ...
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 [] = [] [].
Intersection (set theory) – Set of elements common to all of some sets; Iterated binary operation – Repeated application of an operation to a sequence; List of set identities and relations – Equalities for combinations of sets; Naive set theory – Informal set theories; Symmetric difference – Elements in exactly one of two sets
So if f is a function and x is in its domain, then f ′ x is f(x). f ″ X f ″ X is the image of a set X by f. If f is a function whose domain contains X this is {f(x):x∈X} [ ] 1. M[G] is the smallest model of ZF containing G and all elements of M. 2. [α] β is the set of all subsets of a set α of cardinality β, or of an ordered set α ...
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 algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
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 ...
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 ...