% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
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\newcommand{\MultisetXmpTitle}{Multiset}
\newcommand{\MultisetXmpNumber}{9.57}
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\begin{page}{MultisetXmpPage}{9.57 Multiset}
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The domain \spadtype{Multiset(R)} is similar to \spadtype{Set(R)}
except that multiplicities
(counts of duplications) are maintained and displayed.
Use the operation \spadfunFrom{multiset}{Multiset} to create
%-% \HDindex{set!vs. multiset}{MultisetXmpPage}{9.57}{Multiset}
multisets from lists.
%-% \HDindex{multiset}{MultisetXmpPage}{9.57}{Multiset}
All the standard operations from sets are available for
multisets.
An element with multiplicity greater than one has the
multiplicity displayed first, then a colon, and then the element.

\xtc{
Create a multiset of integers.
}{
\spadpaste{s := multiset [1,2,3,4,5,4,3,2,3,4,5,6,7,4,10]\bound{s}}
}
\xtc{
The operation \spadfunX{insert} adds an element to a multiset.
}{
\spadpaste{insert!(3,s)\bound{s1}\free{s}}
}
\xtc{
Use \spadfunX{remove} to remove an element.
If a third argument is present, it specifies how many instances
to remove. Otherwise all instances of the element are removed.
Display the resulting multiset.
}{
\spadpaste{remove!(3,s,1); s\bound{s2}\free{s1}}
}
\xtc{
}{
\spadpaste{remove!(5,s); s\bound{s2}\free{s1}}
}
\xtc{
The operation \spadfun{count} returns the number of copies
of a given value.
}{
\spadpaste{count(5,s)\free{s2}}
}
\xtc{
A second multiset.
}{
\spadpaste{t := multiset [2,2,2,-9]\bound{t}}
}
\xtc{
The \spadfun{union} of two multisets is additive.
}{
\spadpaste{U := union(s,t)\bound{U}}
}
\xtc{
The \spadfun{intersect} operation gives the elements that are in
common, with additive multiplicity.
}{
\spadpaste{I := intersect(s,t)\bound{I}}
}
\xtc{
The \spadfun{difference} of \spad{s} and \spad{t} consists of
the elements that \spad{s} has but \spad{t} does not.
Elements are regarded as indistinguishable, so that if \spad{s}
and \spad{t} have any element in common, the \spadfun{difference}
does not contain that element.
}{
\spadpaste{difference(s,t)\free{s2 t}}
}
\xtc{
The \spadfun{symmetricDifference} is the \spadfun{union}
of \spad{difference(s, t)} and \spad{difference(t, s)}.
}{
\spadpaste{S := symmetricDifference(s,t)\bound{S}\free{s2 t}}
}
\xtc{
Check that the \spadfun{union} of the \spadfun{symmetricDifference} and
the \spadfun{intersect} equals the \spadfun{union} of the elements.
}{
\spadpaste{(U = union(S,I))@Boolean\free{S I U}}
}
\xtc{
Check some inclusion relations.
}{
\spadpaste{t1 := multiset [1,2,2,3]; [t1 < t, t1 < s, t < s, t1 <= s]\free{t s2}}
}
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