SETS
Acknowledgements
Introduction
Enumerated Sets
Formal Sets
Indexed Sets
Multisets
Compatibility
Notation
Creating Sets
The Formal Set Constructor
The Enumerated Set Constructor
The Indexed Set Constructor
The Multiset Constructor
The Arithmetic Progression Constructors
Power Sets
The Cartesian Product Constructors
Sets from Structures
Accessing and Modifying Sets
Accessing Sets and their Associated Structures
Selecting Elements of Sets
Modifying Sets
Operations on Sets
Boolean Functions and Operators
Binary Set Operators
Other Set Operations
Quantifiers
Reduction and Iteration
Reduction
Iteration
Introduction
Enumerated Sets
Formal Sets
Indexed Sets
Multisets
Compatibility
Notation
Creating Sets
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 (H10E1)
{ 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 (H10E2)
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 (H10E3)
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 (H10E4)
The Arithmetic Progression Constructors
{ i..j } : RngIntElt, RngIntElt -> Set
{ i .. j by k } : RngIntElt, RngIntElt, RngIntElt -> Set
Example Set_Progression (H10E5)
Power Sets
PowerSet(R) : Str -> PowSetEnum
PowerIndexedSet(R) : Str -> PowSetIndx
PowerMultiset(R) : Str -> PowSetMulti
PowerFormalSet(R) : Str -> PowSetIndx
S in P : SetEnum, PowSetEnum -> BoolElt
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 (H10E6)
The Cartesian Product Constructors
Sets from Structures
Set(M) : Str -> SetEnum
FormalSet(M) : Str -> SetFormal
Accessing and Modifying Sets
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 (H10E7)
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 (H10E8)
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 (H10E9)
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 (H10E10)
SetToIndexedSet(E) : SetEnum -> SetIndx
IndexedSetToSet(S) : SetIndx -> SetEnum
IndexedSetToSequence(S) : SetIndx -> SeqEnum
MultisetToSet(S) : SetMulti -> SetEnum
SetToMultiset(E) : SetEnum -> SetMulti
SequenceToMultiset(Q) : SeqEnum -> SetMulti
Operations on Sets
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 (H10E11)
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 (H10E12)
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 (H10E13)
Reduction and Iteration
Reduction
& o S : Op, SetEnum -> Elt
Example Set_Reduction (H10E14)
Iteration
x in S
i -> x in S
x -> v in M
Example Set_Iteration (H10E15)
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