The Formal Set Constructor
{! x in F | P(x) !}
The Enumerated Set Constructor
{ } : Null -> Set
{ U | } : Str -> Set
{ e1, e2, ..., en } : Elt, ..., Elt -> Set
Example Set_Universe (H9E1)
{ U | e1, e2, ..., en } : Str, Elt, ..., Elt -> Set
{ e(x) : x in E | P(x) }
{ U | e(x) : x in E | P(x) }
{ e(x1,...,xk) : x1 in E1, ..., xkin Ek | P(x1, ..., xk) }
{ U | e(x1,...,xk) : x1 in E1, ...,xk in Ek | P(x1, ..., xk) }
Example Set_AlmostFermat (H9E2)
The Indexed Set Constructor
{@ @} : Null -> SetIndx
{@ U | @} : Str -> SetIndx
{@ e1, e2, ..., en @} : Elt, ..., Elt -> SetIndx
{@ U | e1, e2, ..., em @} : Str, Elt, ..., Elt -> SetIndx
{@ e(x) : x in E | P(x) @}
{@ U | e(x) : x in E | P(x) @}
{@ e(x1,...,xk) : x1 in E1, ..., xkin Ek | P(x1, ..., xk) @}
{@ U | e(x1,...,xk) : x1 in E1, ...,xk in Ek | P(x1, ..., xk)@}
Example Set_AlmostFermatIndexed (H9E3)
The Multiset Constructor
{* *} : Null -> SetMulti
{* U | *} : Str -> SetMulti
{* e1, e2, ..., en *} : Elt, ..., Elt -> SetMulti
{* U | e1, e2, ..., em *} : Str, Elt, ..., Elt -> SetMulti
{* e(x) : x in E | P(x) *}
{* U | e(x) : x in E | P(x) *}
{* e(x1,...,xk) : x1 in E1, ..., xkin Ek | P(x1, ..., xk) *}
{* U | e(x1,...,xk) : x1 in E1, ...,xk in Ek | P(x1, ..., xk) *}
Example Set_Multiset (H9E4)
The Arithmetic Progression Constructors
{ i..j } : RngIntElt, RngIntElt -> Set
{ i .. j by k } : RngIntElt, RngIntElt, RngIntElt -> Set
Example Set_Progression (H9E5)
Power Sets
PowerSet(R) : Str -> PowSetEnum
PowerIndexedSet(R) : Str -> PowSetIndx
PowerMultiset(R) : Str -> PowSetMulti
S in P : SetEnum, PowSetEnum -> BoolElt
PowerFormalSet(R) : Str -> PowSetIndx
S in P : SetIndx, PowSetIndx -> BoolElt
S in P : SetMulti, PowSetMulti -> BoolElt
P ! S : PowSetEnum, SetEnum -> SetEnum
P ! S : PowSetIndx, SetIndx -> SetIndx
P ! S : PowSetMulti, SetMulti -> SetMulti
Example Set_PowerSet (H9E6)
The Cartesian Product Constructors
Sets from Structures
Set(M) : Str -> SetEnum
FormalSet(M) : Str -> SetFormal
Accessing Sets and their Associated Structures
# R : SetIndx -> RngIntElt
Category(S) : Any -> Cat
Parent(R) : Set -> Str
Universe(R) : Set -> Str
Index(S, x) : SetIndx, Elt -> RngIntElt
S[i] : SetIndx, RngIntElt -> Elt
S[I] : SetIndx, [RngIntElt] -> SetIndx
Example Set_Miscellaneous (H9E7)
Selecting Elements of Sets
Random(R) : SetIndx -> Elt
random{ e(x) : x in E | P(x) }
random{ e(x1, ..., xk) : x1 in E1,..., xk in Ek | P(x1, ..., xk) }
Example Set_Random (H9E8)
Representative(R) : SetIndx -> Elt
ExtractRep(~R, ~r) : SetEnum, Elt ->
rep{ e(x) : x in E | P(x) }
rep{ e(x1, ..., xk) : x1 in E1, ...,xk in Ek | P(x1, ..., xk) }
Example Set_ExtractRep (H9E9)
Minimum(S) : SetIndx -> Elt, RngIntElt
Maximum(S) : SetIndx -> Elt, RngIntElt
Hash(x) : Elt -> RngIntElt
Modifying Sets
Include(~S, x) : SetEnum, Elt ->
Exclude(~S, x) : SetEnum, Elt ->
ChangeUniverse(~S, V) : SetEnum, Str ->
CanChangeUniverse(S, V) : SetEnum, Str -> Bool, SeqEnum
Example Set_Include (H9E10)
SetToIndexedSet(E) : SetEnum -> SetIndx
IndexedSetToSet(S) : SetIndx -> SetEnum
IndexedSetToSequence(S) : SetIndx -> SeqEnum
MultisetToSet(S) : SetMulti -> SetEnum
SetToMultiset(E) : SetEnum -> SetMulti
SequenceToMultiset(Q) : SeqEnum -> SetMulti
Boolean Functions and Operators
IsNull(R) : SetEnum -> BoolElt
IsEmpty(R) : SetEnum -> BoolElt
x eq y : Elt, Elt -> BoolElt
x ne y : Elt, Elt -> BoolElt
x in R : Elt, Set -> BoolElt
x notin R : Elt, Set -> BoolElt
R subset S : SetEnum, Set -> BoolElt
R notsubset S : SetEnum, Set -> BoolElt
R eq S : Set, Set -> BoolElt
R ne S : Set, Set -> BoolElt
IsDisjoint(R, S) : SetEnum, SetEnum -> BoolElt
Binary Set Operators
R join S : SetEnum, SetEnum -> SetEnum
R meet S : SetEnum, SetEnum -> SetEnum
R diff S : SetEnum, SetEnum -> SetEnum
R sdiff S : SetEnum, SetEnum -> SetEnum
Example Set_Join (H9E11)
Other Set Operations
Multiplicity(S, x) : SetMulti, Elt -> RngIntElt
Multiplicities(S) : SetMulti -> SeqEnum
Subsets(S) : SetEnum -> SetEnum
Subsets(S, k) : SetEnum, RngIntElt -> SetEnum
RandomSubset(S, k) : SetEnum, RngIntElt -> SetEnum
Multisets(S, k) : SetEnum, RngIntElt -> SetEnum
Subsequences(S, k) : SetEnum, RngIntElt -> SetEnum
Permutations(S) : SetEnum -> SetEnum;
Permutations(S, k) : SetEnum, RngIntElt -> SetEnum;
Quantifiers
exists(t){ e(x): x in E | P(x) }
exists(t){ e(x1, ..., xk): x1 in E1,..., xk in Ek | P(x1, ..., xk) }
Example Set_Exists (H9E12)
forall(t){ e(x) : x in E | P(x) }
forall(t){ e(x1, ..., xk): x1 in E1,..., xk in Ek | P(x1, ..., xk) }
Example Set_NestedExists (H9E13)
Reduction and Iteration over Sets
x in S
& o S : Op, SetIndx -> Elt
Example Set_Reduction (H9E14)
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Version: V2.19 of
Mon Dec 17 14:40:36 EST 2012