--- layout: default title: "Sets" parent: Composite ancestor: Funcons-beta --- [Funcons-beta] : [Sets.cbs] ----------------------------- ### Sets 
[
  Type   sets
  Funcon set
  Funcon set-elements
  Funcon is-in-set
  Funcon is-subset
  Funcon set-insert
  Funcon set-unite
  Funcon set-intersect
  Funcon set-difference
  Funcon set-size
  Funcon some-element
  Funcon element-not-in
]
Meta-variables
  GT <: ground-values
Built-in Type
  sets(GT)
sets(GT) is the type of possibly-empty finite sets {V1, ..., Vn} where V1:GT, ..., Vn:GT. 
Built-in Funcon
  set(_:(GT)*) : =>sets(GT)
The notation {V1, ..., Vn} for set(V1, ..., Vn) is built-in. 
Assert
  {V*:(GT)*} == set(V*)
Note that set(...) is not a constructor operation. The order and duplicates of argument values are ignored (e.g., {1,2,1} denotes the same set as {1,2} and {2,1}). 
Built-in Funcon
  set-elements(_:sets(GT)) : =>(GT)*
For each set S, the sequence of values V* returned by set-elements(S) contains each element of S just once. The order of the values in V* is unspecified, and may vary between sets (e.g., set-elements{1,2} could be (1,2) and set-elements{1,2,3} could be (3,2,1)). 
Assert
  set(set-elements(S)) == S
Built-in Funcon
  is-in-set(_:GT, _:sets(GT)) : =>booleans
is-in-set(GV,S) tests whether GV is in the set S. 
Assert
  is-in-set(GV:GT, { }) == false
Assert
  is-in-set(GV:GT, {GV}:sets(GT)) == true
Built-in Funcon
  is-subset(_:sets(GT), _:sets(GT)) : =>booleans
is-subset(S1,S2) tests whether S1 is a subset of S2. 
Assert
  is-subset({ }, S:sets(GT)) == true
Assert
  is-subset(S:sets(GT), S) == true
Built-in Funcon
  set-insert(_:GT, _:sets(GT)) : =>sets(GT)
set-insert(GV, S) returns the set union of {GV} and S. 
Assert
  is-in-set(GV:GT, set-insert(GV:GT, S:sets(GT))) == true
Built-in Funcon
  set-unite(_:(sets(GT))*) : =>sets(GT)
set-unite(...) unites a sequence of sets. 
Assert
  set-unite(S:sets(GT), S) == S
Assert
  set-unite(S1:sets(GT), S2:sets(GT)) == set-unite(S2, S1)
Assert
  set-unite(S1:sets(GT), set-unite(S2:sets(GT), S3:sets(GT))) ==
    set-unite(set-unite(S1, S2), S3)
Assert
  set-unite(S1:sets(GT), S2:sets(GT), S3:sets(GT)) == 
    set-unite(S1, set-unite(S2, S3))
Assert
  set-unite(S:sets(GT)) == S
Assert
  set-unite( ) == { }
Built-in Funcon
  set-intersect(_:(sets(GT))+) : =>sets(GT)
set-intersect(GT,...) intersects a non-empty sequence of sets. 
Assert
  set-intersect(S:sets(GT), S) == S
Assert
  set-intersect(S1:sets(GT), S2:sets(GT)) == set-intersect(S2, S1)
Assert
  set-intersect(S1:sets(GT), set-intersect(S2:sets(GT), S3:sets(GT))) == 
    set-intersect(set-intersect(S1, S2), S3)
Assert
  set-intersect(S1:sets(GT), S2:sets(GT), S3:sets(GT)) == 
    set-intersect(S1, set-intersect(S2, S3))
Assert
  set-intersect(S:sets(GT)) == S
Built-in Funcon
  set-difference(_:sets(GT), _:sets(GT)) : =>sets(GT)
set-difference(S1, S2) returns the set containing those elements of S1 that are not in S2. 
Built-in Funcon
  set-size(_:sets(GT)) : =>natural-numbers
Assert
  set-size(S:sets(GT)) == length(set-elements(S))
Funcon
  some-element(_:sets(GT)) : =>GT?
Assert
  some-element(S:sets(GT)) == index(1, set-elements(S))
Assert
  some-element{ } == ( )
Built-in Funcon
  element-not-in(GT:types, _:set(GT)) : =>GT?
element-not-in(GT, S) gives an element of the type GT not in the set S, or ( ) when S is empty. When the set of elements of GT is infinite, element-not-in(GT, S) never gives ( ). ____ From the [PLanCompS Project] | [CBS-beta issues...] | [Suggest an improvement...] [Sets.cbs]: Sets.cbs "CBS SOURCE FILE" [Funcons-beta]: /CBS-beta/docs/Funcons-beta "FUNCONS-BETA" [Unstable-Funcons-beta]: /CBS-beta/docs/Unstable-Funcons-beta "UNSTABLE-FUNCONS-BETA" [Languages-beta]: /CBS-beta/docs/Languages-beta "LANGUAGES-BETA" [Unstable-Languages-beta]: /CBS-beta/docs/Unstable-Languages-beta "UNSTABLE-LANGUAGES-BETA" [CBS-beta]: /CBS-beta "CBS-BETA" [PLanCompS Project]: https://plancomps.github.io "PROGRAMMING LANGUAGE COMPONENTS AND SPECIFICATIONS PROJECT HOME PAGE" [CBS-beta issues...]: https://github.com/plancomps/CBS-beta/issues "CBS-BETA ISSUE REPORTS ON GITHUB" [Suggest an improvement...]: mailto:plancomps@gmail.com?Subject=CBS-beta%20-%20comment&Body=Re%3A%20CBS-beta%20specification%20at%20Values/Composite/Sets/Sets.cbs%0A%0AComment/Query/Issue/Suggestion%3A%0A%0A%0ASignature%3A%0A "GENERATE AN EMAIL TEMPLATE"