The customary braces { and } are used to define enumerated sets. Formal sets are delimited by the composite braces {! and !}. For indexed sets {@ and @} are used. For multisets {* and *} are used.
The formal set constructor has the following fixed format (the
expressions appearing in the construct are defined above):
{! x in F | P(x) !}
Form the formal set consisting of the subset of elements x of F for which P(x) is true. If P(x) is true for every element of F, the set constructor may be abbreviated to {! x in F !}. Note that the universe of a formal set will always be equal to the carrier set F.
Enumerated sets can be constructed by expressions enclosed in braces,
provided that the values of all expressions can be automatically
coerced into some common structure, as outlined in the Introduction.
All general constructors have an optional universe (U in the
list below) up front, that allows the user to specify into which structure
all terms of the sets should be coerced.
{ } : Null -> Set
The null set: an empty set that does not have its universe defined.
The empty set with universe U.
Given a list of expressions e_1, ..., e_n, defining elements a_1, a_2, ..., a_n all belonging to (or automatically coercible into) a single algebraic structure U, create the set { a_1, a_2, ..., a_n } of elements of U.
> S := { (7^2+1)/5, (8^2+1)/5, (9^2-1)/5 };
> S;
{ 10, 13, 16 }
> Parent(S);
Set of subsets of Rational Field
Thus S was created as a set of rationals, because /
on integers has a rational result. If one wishes to obtain a
set of integers, one could specify the universe (or one could
use div, or one could use ! on every element to coerce
it into the ring of integers):
> T := { Integers() | (7^2+1)/5, (8^2+1)/5, (9^2-1)/5 };
> T;
{ 10, 13, 16 }
> Parent(T);
Set of subsets of Integer Ring
Given a list of expressions e_1, ..., e_n, which define elements a_1, a_2, ..., a_n that are all coercible into U, create the set { a_1, a_2, ..., a_n } of elements of U.
Form the set of elements e(x), all belonging to some common structure, for those x in E with the property that the predicate P(x) is true. The expressions appearing in this construct have the interpretation given in the Introduction (in particular, E must be a finite structure that can be enumerated).If P(x) is true for every value of x in E, then the set constructor may be abbreviated to { e(x) : x in E }.
Form the set of elements of U consisting of the values e(x) for those x in E for which the predicate P(x) is true (an error results if not all e(x) are coercible into U). The expressions appearing in this construct have the same interpretation as before.If P is always true, it may be omitted (including the |).
The set consisting of those elements e(x_1, ..., x_k), in some common structure, for which x_1, ..., x_k in E_1, ..., E_k have the property that P(x_1, ..., x_k) is true. The expressions appearing in this construct have the interpretation given in the Introduction.Note that if two successive allowable structures E_i and E_(i + 1) are identical, then the specification of the carrier sets for x_i and x_(i + 1) may be abbreviated to x_i, x_(i + 1) in E_i.
Also, if P(x_1, ..., x_k) is always true, it may be omitted (including the |).
As in the previous entry, the set consisting of those elements e(x_1, ..., x_k) for which P(x_1, ..., x_k) is true, is formed, as a set of elements of U (an error occurs if not all e(x_1, ..., x_k) are elements of or coercible into U).Again, identical successive structures may be abbreviated, and a predicate that is always true may be omitted.
> cubes := { Integers() | x^3 : x in [1..1000] };
> plus := { <a, b> : a in [2..1000], b in [2..1000] | \
> b ge a and (a^3+b^3-1) in cubes };
> plus;
{
< 9, 10 >,
< 135, 235 >
< 334, 438 >,
< 73, 144 >,
< 64, 94 >,
< 244, 729 >
}
Note that we spend a lot of time cubing integers this way.
For a more efficient approach, see a subsequent example.
The creation of indexed sets is similar to that of enumerated sets.
{@ @} : Null -> SetIndx
The null set: an empty indexed set that does not have its universe defined.
The empty indexed set with universe U.
Given a list of expressions e_1, ..., e_n, defining elements a_1, a_2, ..., a_n all belonging to (or automatically coercible into) a single algebraic structure U, create the indexed set Q = { a_1, a_2, ..., a_n } of elements of U.
Given a list of expressions e_1, ..., e_m, which define elements a_1, a_2, ..., a_n that are all coercible into U, create the indexed set Q = { a_1, a_2, ..., a_n } of elements of U.
Form the indexed set of elements e(x), all belonging to some common structure, for those x in E with the property that the predicate P(x) is true. The expressions appearing in this construct have the interpretation given in the Introduction (in particular, E must be a finite structure that can be enumerated).If P is always true, it may be omitted (including the |).
Form the indexed set of elements of U consisting of the values e(x) for those x in E for which the predicate P(x) is true (an error results if not all e(x) are coercible into U). The expressions appearing in this construct have the same interpretation as before.If P is always true, it may be omitted (including the |).
The indexed set consisting of those elements e(x_1, ..., x_k) (in some common structure), for which x_1, ..., x_k in E_1 x ... x E_k have the property that P(x_1, ..., x_k) is true. The expressions appearing in this construct have the interpretation given in the Introduction.Note that if two successive allowable structures E_i and E_(i + 1) are identical, then the specification of the carrier sets for x_i and x_(i + 1) may be abbreviated to x_i, x_(i + 1) in E_i.
Also, if P(x_1, ..., x_k) is always true, it may be omitted.
As in the previous entry, the indexed set consisting of those elements e(x_1, ..., x_k) for which P(x_1, ..., x_k) is true is formed, as an indexed set of elements of U (an error occurs if not all e(x_1, ..., x_k) are elements of or coercible into U).Again, identical successive structures may be abbreviated, and a predicate that is always true may be omitted.
> cubes := {@ Integers() | z^3 : z in [1..25] @};
> plus := { <x, y, z> : x in [-10..10], y in [-10..10], z in [1..25] |
> y ge x and Abs(x) gt 1 and Abs(y) gt 1 and (x^3+y^3-1) in cubes
> and (x^3+y^3-1) eq cubes[z] };
> plus;
{ <-6, 9, 8>, <9, 10, 12>, <-8, 9, 6> }
The creation of multisets is similar to that of enumerated sets. An
important difference is that repetitions are significant and the
operator ^^ (mentioned above) may be used to specify the multiplicity
of an element.
{* *} : Null -> SetMulti
The null set: an empty multiset that does not have its universe defined.
The empty multiset with universe U.
Given a list of expressions e_1, ..., e_n, defining elements a_1, a_2, ..., a_n all belonging to (or automatically coercible into) a single algebraic structure U, create the multiset Q = {* a_1, a_2, ..., a_n *} of elements of U.
Given a list of expressions e_1, ..., e_m, which define elements a_1, a_2, ..., a_n that are all coercible into U, create the multiset Q = {* a_1, a_2, ..., a_n *} of elements of U.
Form the multiset of elements e(x), all belonging to some common structure, for those x in E with the property that the predicate P(x) is true. The expressions appearing in this construct have the interpretation given in the Introduction (in particular, E must be a finite structure that can be enumerated).If P is always true, it may be omitted (including the |).
Form the multiset of elements of U consisting of the values e(x) for those x in E for which the predicate P(x) is true (an error results if not all e(x) are coercible into U). The expressions appearing in this construct have the same interpretation as before.If P is always true, it may be omitted (including the |).
The multiset consisting of those elements e(x_1, ..., x_k) (in some common structure), for which x_1, ..., x_k in E_1 x ... x E_k have the property that P(x_1, ..., x_k) is true. The expressions appearing in this construct have the interpretation given in the Introduction.Note that if two successive allowable structures E_i and E_(i + 1) are identical, then the specification of the carrier sets for x_i and x_(i + 1) may be abbreviated to x_i, x_(i + 1) in E_i.
Also, if P(x_1, ..., x_k) is always true, it may be omitted.
As in the previous entry, the multiset consisting of those elements e(x_1, ..., x_k) for which P(x_1, ..., x_k) is true is formed, as an multiset of elements of U (an error occurs if not all e(x_1, ..., x_k) are elements of or coercible into U).Again, identical successive structures may be abbreviated, and a predicate that is always true may be omitted.
> M := {* 1, 1, 1, 3, 5 *};
> M;
{* 1^^3, 3, 5 *}
> M := {* 1^^4, 2^^5, 1/2^^3 *};
> M;
> // Count frequency of digits in first 1000 digits of pi:
> pi := Pi(RealField(1001));
> dec1000 := Round(10^1000*(pi-3));
> I := IntegerToString(dec1000);
> F := {* I[i]: i in [1 .. #I] *};
> F;
{* 7^^95, 3^^102, 6^^94, 2^^103, 9^^106, 5^^97,
1^^116, 8^^101, 4^^93, 0^^93 *}
> for i := 0 to 9 do print i, Multiplicity(F, IntegerToString(i)); end for;
0 93
1 116
2 103
3 102
4 93
5 97
6 94
7 95
8 101
9 106
Some special constructors exist to create and store enumerated
sets of integers in arithmetic progression efficiently.
This only works for arithmetic progressions of elements of
the ring of integers.
{ i..j } : RngIntElt, RngIntElt -> Set
The enumerated set whose elements form the arithmetic progression i, i + 1, i + 2, ..., j, where i and j are (expressions defining) integers. If j is less than i then the empty set will be created.The only universe U that is legal here is the ring of integers.
The enumerated set consisting of the integers forming the arithmetic progression i, i + k, i + 2 * k, ..., j, where i, j and k are (expressions defining) integers (but k != 0).If k is positive then the last element in the progression will be the greatest integer of the form i + n * k that is less than or equal to j. If j is less than i, the empty set will be constructed.
If k is negative then the last element in the progression will be the least integer of the form i + n * k that is greater than or equal to j. If j is greater than i, the empty set will be constructed.
As for the previous constructor, only the ring of integers is allowed as a legal universe U.
> S := { FiniteField(13) | 1..10 }; Runtime error in { .. }: Invalid set universe > S := { FiniteField(13) | x : x in { 1..10 } }; > S; { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } > G := PowerSet(FiniteField(13)); > S := G ! { 1..10 }; > S; { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }