Enumerated sets can be modified by inserting or removing elements. Indexed sets allow some sequence-like operators for modification and access.
Cardinality of the enumerated, indexed, or multi- set R. Note that for a multiset, repetitions are significant, so the result may be greater than the underlying set.
The category of the object S. For a set this will be one of SetEnum, SetIndx, SetMulti, or SetFormal. For a power set the type is one of PowSetEnum, PowSetIndx, PowSetMulti.
Returns the parent structure of R, that is, the structure consisting of all (enumerated) sequences over the universe of R.
Returns the `universe' of the (enumerated or indexed or multi- or formal) set R, that is, the common structure to which all elements of the set belong. An error is signalled when R is the null set.
Given an indexed set S, and an element x, returns the index i such that S[i]=x if such index exists, or return 0 if x is not in S. If x is not in the universe of S, an attempt will be made to coerce it; an error occurs if this fails.
Return the i-th entry of indexed set S. If i < 1 or i > #S an error occurs. Note that indexing is not allowed on the left hand side.
The indexed set {S[i1], ..., S[ir]} consisting of terms selected from the indexed set S, according to the terms of the integer sequence I. If any term of I lies outside the range 1 to #S, then an error results. If I is the empty sequence, then the empty set with universe the same as that of S is returned.
> B := {@ { i : i in [1..k] } : k in [1..5] @};
> B;
{@
{ 1 },
{ 1, 2 },
{ 1, 2, 3 },
{ 1, 2, 3, 4 },
{ 1, 2, 3, 4, 5 },
@}
> #B;
5
> Universe(B);
Set of subsets of Integer Ring
> Parent(B);
Set of indexed subsets of Set of subsets of Integer Ring
> Category(B);
SetIndx
> Index(B, { 2, 1 });
2
> #B[2];
2
> Universe(B[2]);
Integer Ring
Most finite structures in Magma, including enumerated sets, allow one to obtain a random element using Random. There is an alternative (and often preferable) option for enumerated sets in the random{ } constructor. This makes it possible to choose a random element of the set without generating the whole set first.
Likewise, rep{ } is an alternative to the general Rep function returning a representative element of a structure, having the advantage of aborting the construction of the set as soon as one element has been found.
Here, E will again be an enumerable structure, that is, a structure that allows enumeration of its elements (see the Appendix for an exhaustive list).
Note that random{ e(x) : x in E | P(x) } does not return a random element of the set of values e(x), but rather a value of e(x) for a random x in E which satisfies P (and mutatis mutandis for rep).
See the subsection on Notation in the Introduction (Chapter INTRODUCTION TO AGGREGATES) for conventions regarding e, x, E, P.
A random element chosen from the enumerated, indexed or multi- set R. Every element has an equal probability of being chosen for enumerated or indexed sets, and a weighted probability in proportion to its multiplicity for multisets. Successive invocations of the function will result in independently chosen elements being returned as the value of the function. If R is empty an error occurs.
Given an enumerated structure E and a Boolean expression P, return the value of the expression e(y) for a randomly chosen element y of E for which P(y) is true.The expression P may be omitted if it is always true.
Given enumerated structures E1, ..., Ek, and a Boolean expression P(x1, ..., xk), return the value of the expression e(y1, ..., yk) for a randomly chosen element < y1, ..., yk > of E1 x ... x Ek, for which P(y1, ..., yk) is true.The expression P may be omitted if it is always true.
If successive structures Ei and Ei + 1 are identical, then the abbreviation xi, xi + 1 in Ei may be used.
> p := 10007;
> F := FiniteField(p);
> proots := { z : z in F | IsPrimitive(z) };
> #proots;
5002
> Random(proots);
5279
This way, a set of 5002 elements is built (and primitivity is checked
for all elements of F), and a random choice is made.
Alternatively, we use random.
> random{ x : x in F | IsPrimitive(x) };
4263
In this case random elements in F are chosen until one is found that
is primitive. Since almost half of F's elements are primitive, only
very few primitivity tests will be done before success occurs.
An arbitrary element chosen from the enumerated, indexed, or multi- set R.
Assigns an arbitrary element chosen from the enumerated set R to r, and removes it from R. Thus the set R is modified, as well as the element r. An error occurs if R is empty.
Given an enumerated structure E and a Boolean expression P, return the value of the expression e(y) for the first element y of E for which P(y) is true. If P(x) is false for every element of E, an error will occur.
Given enumerated structures E1, ..., Ek, and a Boolean expression P(x1, ..., xk), return the value of the expression e(y1, ..., yk) for the first element < y1, ..., yk > of E1 x ... x Ek, for which P(y1, ..., yk) is true. An error occurs if no element of E1 x ... x Ek satisfies P.The expression P may be omitted if it is always true.
If successive structures Ei and Ei + 1 are identical, then the abbreviation xi, xi + 1 in Ei may be used.
> cubes := { Integers() | x^3 : x in [1..1000] };
> cc := cubes;
> min := { };
> while not IsEmpty(cc) do
> ExtractRep(~cc, ~a);
> for b in cc do
> if a+b+1 in cubes then
> min join:= { <a, b> };
> end if;
> end for;
> end while;
> { < Iroot(x[1], 3), Iroot(x[2], 3) > : x in min };
{ <138, 135>, <823, 566>, <426, 372>, <242, 720>,
<138, 71>, <426, 486>, <6, 8> }
Note that instead of taking cubes over again, we only have to take
cube roots in the last line (on the small resulting set) once.
Given a non-empty enumerated, indexed, or multi- set S, such that lt and eq are defined on the universe of S, this function returns the minimum of the elements of S. If S is an indexed set, the position of the minimum is also returned.
Given a non-empty enumerated, indexed, or multi- set S, such that lt and eq are defined on the universe of S, this function returns the maximum of the elements of S. If S is an indexed set, the position of the maximum is also returned.
Given a Magma object x which can be placed in a set, return the hash value of x used by the set machinery. This is a fixed but arbitrary non-negative integer (whose maximum value is the maximum value of a C unsigned long on the particular machine). The crucial property is that if x and y are objects and x equals y then the hash values of x and y are equal (even if x and y have different internal structures). Thus one could implement sets manually if desired by the use of this function.
Create the enumerated, indexed, or multi- set obtained by putting the element x in S (S is unchanged if S is not a multiset and x is already in S). If S is an indexed set, the element will be appended at the end. If S is a multiset, the multiplicity of x will be increased accordingly. If x is not in the universe of S, an attempt will be made to coerce it; an error occurs if this fails.There are two versions of this: a procedure, where S is replaced by the new set, and a function, which returns the new set. The procedural version takes a reference ~S to S as an argument.
Note that the procedural version is much more efficient since the set S will not be copied.
Create a new set by removing the element x from S. If S is an enumerated set, nothing happens if x is not in S. If S is a multiset, the multiplicity of x will be decreased accordingly. If x is not in the universe of S, an attempt will be made to coerce it; an error occurs if this fails.There are two versions of this: a procedure, where S is replaced by the new set, and a function, which returns the new set. The procedural version takes a reference ~S to S as an argument.
Note that the procedural version is much more efficient since the set S will not be copied.
Given an enumerated, indexed, or multi- set S with universe U and a structure V which contains U, construct a new set of the same type which consists of the elements of S coerced into V.There are two versions of this: a procedure, where S is replaced by the new set, and a function, which returns the new set. The procedural version takes a reference ~S to S as an argument.
Note that the procedural version is much more efficient since the set S will not be copied.
Given an enumerated, indexed, or multi- set S with universe U and a structure V which contains U, attempt to construct a new set T of the same type which consists of the elements of S coerced into V; if successful, return true and T, otherwise return false.
> R := { 218, 271, 511 };
> x := 0;
> cubes := { 0 };
> while not IsEmpty(R) do
> x +:= 1;
> c := x^3;
> Include(~cubes, c);
> Include(~cubes, -c);
> for z in cubes do
> Exclude(~R, z+c);
> Exclude(~R, z-c);
> end for;
> end while;
We did not record how the elements of R were obtained as sums of a pair
of cubes. For that, the following suffices.
> R := { 218, 271, 511 }; // it has been emptied !
> { { x, y } : x, y in cubes | x+y in R };
{
{ -729, 1000 },
{ -125, 343 },
{ -1, 512 },
}
Given an enumerated set E, this function returns an indexed set with the same elements (and universe) as E.
Given an indexed set S, this function returns an enumerated set with the same elements (and universe) as E.
Given an indexed set S, this function returns a sequence with the same elements (and universe) as E.
Given a multiset S, this function returns an enumerated set with the same elements (and universe) as S.
Given an enumerated set E, this function returns a multiset with the same elements (and universe) as E.
Given an enumerated sequence E, this function returns a multiset with the same elements (and universe) as E.