MutableSet - Maple Help

MutableSet

An object-based mutable set

Description

 • The MutableSet appliable object provides an efficient means to construct, manipulate, and query set objects.
 • Unlike a standard Maple set, adding an element to a MutableSet object does not allocate a new container.  As such, it can be efficiently used in loops to build up a set an element at a time.

Construction

 • The MutableSet function returns a new MutableSet object.
 – If called with a single argument that is either a Maple set or a MutableSet object, the returned object contains the elements in the argument.
 – Otherwise, the returned object contains each distinct argument as an element.

Modification

 • The following functions modify the content The ms argument denotes a MutableSet object.
 – clear(ms) : Clear ms.
 – delete(ms,expr) : Delete expr from ms.
 – insert(ms,expr) : Insert expr into ms.
 – ms ,= expr : Insert expr into ms. Unlike the insert function, the ,= operator can accept a sequence of elements on the right hand side.

Set Operation

 • The following set functions combine MutableSet objects. Those prefixed with an ampersand (&) operate inplace on the first argument. These functions can be invoked as functions or inline operators. The ms1 and ms2 arguments denote MutableSet objects.
 – ms1 intersect ms2, or $\mathrm{ms1}\cap \mathrm{ms2}$, or intersect(ms1,ms2,...) : Return a new MutableSet object that is the intersection of the given MutableSets.
 – ms1 &intersect ms2 or &intersect(ms1,ms2,...) : Remove elements of ms1 that are not elements of ms2, etc., updating ms1.
 – ms1 minus ms2, or $\mathrm{ms1}\setminus \mathrm{ms2}$, or minus(ms1,ms2) : Return a new MutableSet object that contains the set difference of ms1 and ms2.
 – ms1 &minus ms2 or &minus(ms1,ms2) : Remove elements of ms2 from ms1, updating ms1.
 – ms1 union ms2, or $\mathrm{ms1}\cup \mathrm{ms2}$, or union(ms1,ms2,...) : Return a new MutableSet object that is the union of the given mutable sets.
 – ms1 &union ms2 or &union(ms1,ms2,...) : Combine the members of ms2, etc., with ms1, updating ms1.

Inspection

 • The following functions inspect the content of a MutableSet object. The ms, ms1, and ms2 arguments denote MutableSet objects.
 – ms = other or =(ms,other) : When evaluated in a boolean context, returns true if ms and other are both MutableSet objects with identical content, false otherwise.
 – expr in ms  or in(expr,ms) : Returns true if expr is a member of ms, false otherwise.
 – empty(ms) : Returns true if ms is empty, false otherwise.
 – entries(ms) : Returns an expression sequence of all the entries in ms, with each entry enclosed in a list by default.
 – has(ms,expr) : Returns true if any member of ms contains expr, false otherwise.
 – hastype(ms,typ) : Returns true if any member of ms contains an expression of the specified type, false otherwise.
 – indets(ms,typ) : Returns a Maple set of all indeterminates in ms. If the optional typ parameter is specified, then returns the expressions of type typ.
 – indices(ms) : Returns an expression sequence of all the indices of ms, with each index enclosed in a list by default.
 – lowerbound(ms) : Returns the lower index bound (always 1).
 – max(ms) : Returns the maximum element of ms. Behaves like max on sets.
 – member(expr,ms) : Returns true if  expr is a member of ms, false otherwise.  The three argument form of member is not supported.
 – min(ms) : Returns the minimum element of ms.  Behaves like min on sets.
 – numelems(ms) : Returns the number of elements in ms.
 – numboccur(ms,expr) : Count the number of occurrences of expr in ms, either as an element or within elements.
 – subset(ms1,ms2), or ms1 subset ms2, or ms1 subset ms2 : Returns true if ms1 is a subset of ms2, false otherwise.
 – upperbound(ms) : Returns the upper index (same as the output of numelems).
 • Each of the above functions is already available in Maple for built-in Maple structures such as sets and Arrays. Please refer to the help page for each function for details on usage.

Mapping, Selection, Removal, and Substitution

 • The map, select, remove, selectremove, and subs functions operate on a MutableSet object. Refer to their help pages for the calling sequences. By default they create a new MutableSet object, but if called with the inplace index option, they update the MutableSet object. The ms argument denotes a MutableSet object.
 – map(fcn,ms) : Apply a procedure, fcn, to each element of ms.
 – select(fcn,ms) : Select elements of ms for which fcn returns true.
 – remove(fcn,ms) : Remove elements of ms for which fcn returns true.
 – selectremove(fcn,dq) : Produce two MutableSets, one containing selected elements and the other removed elements.
 – subs(eqns,ms) : Substitute subexpressions in the content of ms.

Conversion

 • The convert function can be used to convert a MutableSet object to and from other Maple structures.
 – convert(ms,typ) : Convert a MutableSet to the specified type
 – convert(expr,MutableSet) : Convert expr to a MutableSet object. expr must be a list, set, or rtable.

Indexing

 • Integer indices can be used to extract or reassign a single element of a MutableSet. Reassigning an element may change its index.
 • Range indices are not supported.

Iteration

 • The MutableSet object exports a ModuleIterator function, allowing a MutableSet object to be used in loops and iterations (seq, add, etc.).

Examples

Construction

 • Create a MutableSet with three elements.
 > $\mathrm{M1}≔\mathrm{MutableSet}\left(a,b,c\right)$
 ${\mathrm{M1}}{≔}{\mathrm{MutableSet}}{}\left({a}{,}{b}{,}{c}\right)$ (1)
 • Copy M1.
 > $\mathrm{M2}≔\mathrm{MutableSet}\left(\mathrm{M1}\right)$
 ${\mathrm{M2}}{≔}{\mathrm{MutableSet}}{}\left({a}{,}{b}{,}{c}\right)$ (2)
 • Create a MutableSet from a Maple set.
 > $\mathrm{M3}≔\mathrm{MutableSet}\left(\left\{b,c,d\right\}\right)$
 ${\mathrm{M3}}{≔}{\mathrm{MutableSet}}{}\left({b}{,}{c}{,}{d}\right)$ (3)

Modification

 • Delete the b element from M1 and insert a d element.
 > $\mathrm{delete}\left(\mathrm{M1},b\right)$
 ${\mathrm{MutableSet}}{}\left({a}{,}{c}\right)$ (4)
 > $\mathrm{insert}\left(\mathrm{M1},d\right)$
 ${\mathrm{MutableSet}}{}\left({a}{,}{c}{,}{d}\right)$ (5)
 • Add $c$ to each element in $\mathrm{M2}$, doing so inplace.
 > ${\mathrm{map}}_{\mathrm{inplace}}\left(\mathrm{+},\mathrm{M2},c\right)$
 ${\mathrm{MutableSet}}{}\left({a}{+}{c}{,}{b}{+}{c}{,}{2}{}{c}\right)$ (6)
 • Replace all occurrences of $c$ inside $\mathrm{M2}$ with $d$, doing so inplace.
 > ${\mathrm{subs}}_{\mathrm{inplace}}\left(c=d,\mathrm{M2}\right)$
 ${\mathrm{MutableSet}}{}\left({2}{}{d}{,}{a}{+}{d}{,}{b}{+}{d}\right)$ (7)

Operation

 • Create two MutableSet objects, then form their intersection.
 > $\mathrm{M1}≔\mathrm{MutableSet}\left(a,b,c,d\right)$
 ${\mathrm{M1}}{≔}{\mathrm{MutableSet}}{}\left({a}{,}{b}{,}{c}{,}{d}\right)$ (8)
 > $\mathrm{M2}≔\mathrm{MutableSet}\left(c,d,e,f\right)$
 ${\mathrm{M2}}{≔}{\mathrm{MutableSet}}{}\left({c}{,}{d}{,}{e}{,}{f}\right)$ (9)
 > $\mathrm{M3}≔\mathrm{M1}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}∩\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{M2}$
 ${\mathrm{M3}}{≔}{\mathrm{MutableSet}}{}\left({c}{,}{d}\right)$ (10)
 • Use the inplace form to update $\mathrm{M1}$ with the intersection of $\mathrm{M1}$ and $\mathrm{M2}$.
 > $\mathrm{M1}&intersect\mathrm{M2}:$
 > $\mathrm{M1}$
 ${\mathrm{MutableSet}}{}\left({c}{,}{d}\right)$ (11)

Inspection

 • Use member to determine whether a given element is a member of $\mathrm{M1}$.
 > $\mathrm{M1}≔\mathrm{MutableSet}\left(a,b,c,4\right)$
 ${\mathrm{M1}}{≔}{\mathrm{MutableSet}}{}\left({4}{,}{a}{,}{b}{,}{c}\right)$ (12)
 > $\mathrm{member}\left(b,\mathrm{M1}\right)$
 ${\mathrm{true}}$ (13)
 • Use has to test for the occurrence of a subexpression in any of the members of the set.
 > $\mathrm{has}\left(\mathrm{M1},c\right)$
 ${\mathrm{true}}$ (14)
 > $\mathrm{indets}\left(\mathrm{M1},\mathrm{numeric}\right)$
 $\left\{{4}\right\}$ (15)

Conversion

 • Create a MutableSet object by converting a Vector.
 > $\mathrm{M1}≔\mathrm{convert}\left(\mathrm{Vector}\left(10,\mathrm{symbol}=v\right),\mathrm{MutableSet}\right)$
 ${\mathrm{M1}}{≔}{\mathrm{MutableSet}}{}\left({{v}}_{{1}}{,}{{v}}_{{2}}{,}{{v}}_{{3}}{,}{{v}}_{{4}}{,}{{v}}_{{5}}{,}{{v}}_{{6}}{,}{{v}}_{{7}}{,}{{v}}_{{8}}{,}{{v}}_{{9}}{,}{{v}}_{{10}}\right)$ (16)
 • Convert M1 to a standard set.
 > $\mathrm{convert}\left(\mathrm{M1},\mathrm{set}\right)$
 $\left\{{{v}}_{{1}}{,}{{v}}_{{2}}{,}{{v}}_{{3}}{,}{{v}}_{{4}}{,}{{v}}_{{5}}{,}{{v}}_{{6}}{,}{{v}}_{{7}}{,}{{v}}_{{8}}{,}{{v}}_{{9}}{,}{{v}}_{{10}}\right\}$ (17)

Indexing

 • Use indexing to extract each element of a MutableSet.
 > $\mathrm{M1}≔\mathrm{MutableSet}\left(a,b,c,d\right)$
 ${\mathrm{M1}}{≔}{\mathrm{MutableSet}}{}\left({a}{,}{b}{,}{c}{,}{d}\right)$ (18)
 > $\mathrm{seq}\left(i={\mathrm{M1}}_{i},i=1..\mathrm{numelems}\left(\mathrm{M1}\right)\right)$
 ${1}{=}{a}{,}{2}{=}{b}{,}{3}{=}{c}{,}{4}{=}{d}$ (19)
 • Reassign the value at the second index. Note that in the updated MutableSet, the new value is assigned a different index.
 > ${\mathrm{M1}}_{2}≔23:$
 > $\mathrm{seq}\left(i={\mathrm{M1}}_{i},i=1..\mathrm{numelems}\left(\mathrm{M1}\right)\right)$
 ${1}{=}{23}{,}{2}{=}{a}{,}{3}{=}{c}{,}{4}{=}{d}$ (20)

Iteration

 • Iterate through each element in M1.
 > $\left[\mathrm{seq}\left(x,x\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}∈\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{M1}\right)\right]$
 $\left[{23}{,}{a}{,}{c}{,}{d}\right]$ (21)
 • Add the elements of M2.
 > $\mathrm{add}\left(x,x\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}∈\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{M2}\right)$
 ${c}{+}{d}{+}{e}{+}{f}$ (22)
 > $\mathbf{for}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}x\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{M2}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{print}\left(x\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end do}$
 ${c}$
 ${d}$
 ${e}$
 ${f}$ (23)

Usage

 • The MutableSet object provides an efficient way to construct a set an element at a time, in a loop.  Doing so with standard Maple sets is inefficient in that each addition creates a new set. As such the operation is $\mathrm{O}\left({n}^{2}\right)$ in time and memory, where $n$ is the number of elements. The following compares the memory and time required to create a set of ten thousand elements using standard sets and a MutableSet object.
 > MapleSets := proc(n :: posint)  local i, s;     s := {};     for i to n do         s := s union {i};     end do:     numelems(s); end proc: MutableSets := proc(n :: posint) local i, s;     s := MutableSet();     for i to n do         insert(s,-i);     end do;     numelems(s); end proc:
 > $\mathrm{CodeTools}:-\mathrm{Usage}\left(\mathrm{MapleSets}\left(10000\right)\right)$
 memory used=382.91MiB, alloc change=25.47MiB, cpu time=1.69s, real time=1.35s, gc time=714.98ms
 ${10000}$ (24)
 > $\mathrm{CodeTools}:-\mathrm{Usage}\left(\mathrm{MutableSets}\left(10000\right)\right)$
 memory used=9.38MiB, alloc change=8.00MiB, cpu time=184.00ms, real time=119.00ms, gc time=132.99ms
 ${10000}$ (25)

Compatibility

 • The MutableSet object was introduced in Maple 2016.