\section{\module{sets} ---
         Unordered collections of unique elements}

\declaremodule{standard}{sets}
\modulesynopsis{Implementation of sets of unique elements.}
\moduleauthor{Greg V. Wilson}{gvwilson@nevex.com}
\moduleauthor{Alex Martelli}{aleax@aleax.it}
\moduleauthor{Guido van Rossum}{guido@python.org}
\sectionauthor{Raymond D. Hettinger}{python@rcn.com}

\versionadded{2.3}

The \module{sets} module provides classes for constructing and manipulating
unordered collections of unique elements.  Common uses include membership
testing, removing duplicates from a sequence, and computing standard math
operations on sets such as intersection, union, difference, and symmetric
difference.

Like other collections, sets support \code{\var{x} in \var{set}},
\code{len(\var{set})}, and \code{for \var{x} in \var{set}}.  Being an
unordered collection, sets do not record element position or order of
insertion.  Accordingly, sets do not support indexing, slicing, or
other sequence-like behavior.

Most set applications use the \class{Set} class which provides every set
method except for \method{__hash__()}. For advanced applications requiring
a hash method, the \class{ImmutableSet} class adds a \method{__hash__()}
method but omits methods which alter the contents of the set. Both
\class{Set} and \class{ImmutableSet} derive from \class{BaseSet}, an
abstract class useful for determining whether something is a set:
\code{isinstance(\var{obj}, BaseSet)}.

The set classes are implemented using dictionaries.  Accordingly, the
requirements for set elements are the same as those for dictionary keys;
namely, that the element defines both \method{__eq__} and \method{__hash__}.
As a result, sets
cannot contain mutable elements such as lists or dictionaries.
However, they can contain immutable collections such as tuples or
instances of \class{ImmutableSet}.  For convenience in implementing
sets of sets, inner sets are automatically converted to immutable
form, for example, \code{Set([Set(['dog'])])} is transformed to
\code{Set([ImmutableSet(['dog'])])}.

\begin{classdesc}{Set}{\optional{iterable}}
Constructs a new empty \class{Set} object.  If the optional \var{iterable}
parameter is supplied, updates the set with elements obtained from iteration.
All of the elements in \var{iterable} should be immutable or be transformable
to an immutable using the protocol described in
section~\ref{immutable-transforms}.
\end{classdesc}

\begin{classdesc}{ImmutableSet}{\optional{iterable}}
Constructs a new empty \class{ImmutableSet} object.  If the optional
\var{iterable} parameter is supplied, updates the set with elements obtained
from iteration.  All of the elements in \var{iterable} should be immutable or
be transformable to an immutable using the protocol described in
section~\ref{immutable-transforms}.

Because \class{ImmutableSet} objects provide a \method{__hash__()} method,
they can be used as set elements or as dictionary keys.  \class{ImmutableSet}
objects do not have methods for adding or removing elements, so all of the
elements must be known when the constructor is called.
\end{classdesc}


\subsection{Set Objects \label{set-objects}}

Instances of \class{Set} and \class{ImmutableSet} both provide
the following operations:

\begin{tableiii}{c|c|l}{code}{Operation}{Equivalent}{Result}
  \lineiii{len(\var{s})}{}{cardinality of set \var{s}}

  \hline
  \lineiii{\var{x} in \var{s}}{}
         {test \var{x} for membership in \var{s}}
  \lineiii{\var{x} not in \var{s}}{}
         {test \var{x} for non-membership in \var{s}}
  \lineiii{\var{s}.issubset(\var{t})}{\code{\var{s} <= \var{t}}}
         {test whether every element in \var{s} is in \var{t}}
  \lineiii{\var{s}.issuperset(\var{t})}{\code{\var{s} >= \var{t}}}
         {test whether every element in \var{t} is in \var{s}}

  \hline
  \lineiii{\var{s}.union(\var{t})}{\var{s} \textbar{} \var{t}}
         {new set with elements from both \var{s} and \var{t}}
  \lineiii{\var{s}.intersection(\var{t})}{\var{s} \&\ \var{t}}
         {new set with elements common to \var{s} and \var{t}}
  \lineiii{\var{s}.difference(\var{t})}{\var{s} - \var{t}}
         {new set with elements in \var{s} but not in \var{t}}
  \lineiii{\var{s}.symmetric_difference(\var{t})}{\var{s} \^\ \var{t}}
         {new set with elements in either \var{s} or \var{t} but not both}
  \lineiii{\var{s}.copy()}{}
         {new set with a shallow copy of \var{s}}
\end{tableiii}

Note, the non-operator versions of \method{union()},
\method{intersection()}, \method{difference()}, and
\method{symmetric_difference()} will accept any iterable as an argument.
In contrast, their operator based counterparts require their arguments to
be sets.  This precludes error-prone constructions like
\code{Set('abc') \&\ 'cbs'} in favor of the more readable
\code{Set('abc').intersection('cbs')}.
\versionchanged[Formerly all arguments were required to be sets]{2.3.1}

In addition, both \class{Set} and \class{ImmutableSet}
support set to set comparisons.  Two sets are equal if and only if
every element of each set is contained in the other (each is a subset
of the other).
A set is less than another set if and only if the first set is a proper
subset of the second set (is a subset, but is not equal).
A set is greater than another set if and only if the first set is a proper
superset of the second set (is a superset, but is not equal).

The subset and equality comparisons do not generalize to a complete
ordering function.  For example, any two disjoint sets are not equal and
are not subsets of each other, so \emph{all} of the following return
\code{False}:  \code{\var{a}<\var{b}}, \code{\var{a}==\var{b}}, or
\code{\var{a}>\var{b}}.
Accordingly, sets do not implement the \method{__cmp__} method.

Since sets only define partial ordering (subset relationships), the output
of the \method{list.sort()} method is undefined for lists of sets.

The following table lists operations available in \class{ImmutableSet}
but not found in \class{Set}:

\begin{tableii}{c|l}{code}{Operation}{Result}
  \lineii{hash(\var{s})}{returns a hash value for \var{s}}
\end{tableii}

The following table lists operations available in \class{Set}
but not found in \class{ImmutableSet}:

\begin{tableiii}{c|c|l}{code}{Operation}{Equivalent}{Result}
  \lineiii{\var{s}.update(\var{t})}
         {\var{s} \textbar= \var{t}}
         {return set \var{s} with elements added from \var{t}}
  \lineiii{\var{s}.intersection_update(\var{t})}
         {\var{s} \&= \var{t}}
         {return set \var{s} keeping only elements also found in \var{t}}
  \lineiii{\var{s}.difference_update(\var{t})}
         {\var{s} -= \var{t}}
         {return set \var{s} after removing elements found in \var{t}}
  \lineiii{\var{s}.symmetric_difference_update(\var{t})}
         {\var{s} \textasciicircum= \var{t}}
         {return set \var{s} with elements from \var{s} or \var{t}
          but not both}

  \hline
  \lineiii{\var{s}.add(\var{x})}{}
         {add element \var{x} to set \var{s}}
  \lineiii{\var{s}.remove(\var{x})}{}
         {remove \var{x} from set \var{s}; raises \exception{KeyError}
	  if not present}
  \lineiii{\var{s}.discard(\var{x})}{}
         {removes \var{x} from set \var{s} if present}
  \lineiii{\var{s}.pop()}{}
         {remove and return an arbitrary element from \var{s}; raises
	  \exception{KeyError} if empty}
  \lineiii{\var{s}.clear()}{}
         {remove all elements from set \var{s}}
\end{tableiii}

Note, the non-operator versions of \method{update()},
\method{intersection_update()}, \method{difference_update()}, and
\method{symmetric_difference_update()} will accept any iterable as
an argument.
\versionchanged[Formerly all arguments were required to be sets]{2.3.1}

Also note, the module also includes a \method{union_update()} method
which is an alias for \method{update()}.  The method is included for
backwards compatibility.  Programmers should prefer the
\method{update()} method because it is supported by the builtin
\class{set()} and \class{frozenset()} types.

\subsection{Example \label{set-example}}

\begin{verbatim}
>>> from sets import Set
>>> engineers = Set(['John', 'Jane', 'Jack', 'Janice'])
>>> programmers = Set(['Jack', 'Sam', 'Susan', 'Janice'])
>>> managers = Set(['Jane', 'Jack', 'Susan', 'Zack'])
>>> employees = engineers | programmers | managers           # union
>>> engineering_management = engineers & managers            # intersection
>>> fulltime_management = managers - engineers - programmers # difference
>>> engineers.add('Marvin')                                  # add element
>>> print engineers
Set(['Jane', 'Marvin', 'Janice', 'John', 'Jack'])
>>> employees.issuperset(engineers)           # superset test
False
>>> employees.union_update(engineers)         # update from another set
>>> employees.issuperset(engineers)
True
>>> for group in [engineers, programmers, managers, employees]:
...     group.discard('Susan')                # unconditionally remove element
...     print group
...
Set(['Jane', 'Marvin', 'Janice', 'John', 'Jack'])
Set(['Janice', 'Jack', 'Sam'])
Set(['Jane', 'Zack', 'Jack'])
Set(['Jack', 'Sam', 'Jane', 'Marvin', 'Janice', 'John', 'Zack'])
\end{verbatim}


\subsection{Protocol for automatic conversion to immutable
            \label{immutable-transforms}}

Sets can only contain immutable elements.  For convenience, mutable
\class{Set} objects are automatically copied to an \class{ImmutableSet}
before being added as a set element.

The mechanism is to always add a hashable element, or if it is not
hashable, the element is checked to see if it has an
\method{__as_immutable__()} method which returns an immutable equivalent.

Since \class{Set} objects have a \method{__as_immutable__()} method
returning an instance of \class{ImmutableSet}, it is possible to
construct sets of sets.

A similar mechanism is needed by the \method{__contains__()} and
\method{remove()} methods which need to hash an element to check
for membership in a set.  Those methods check an element for hashability
and, if not, check for a \method{__as_temporarily_immutable__()} method
which returns the element wrapped by a class that provides temporary
methods for \method{__hash__()}, \method{__eq__()}, and \method{__ne__()}.

The alternate mechanism spares the need to build a separate copy of
the original mutable object.

\class{Set} objects implement the \method{__as_temporarily_immutable__()}
method which returns the \class{Set} object wrapped by a new class
\class{_TemporarilyImmutableSet}.

The two mechanisms for adding hashability are normally invisible to the
user; however, a conflict can arise in a multi-threaded environment
where one thread is updating a set while another has temporarily wrapped it
in \class{_TemporarilyImmutableSet}.  In other words, sets of mutable sets
are not thread-safe.


\subsection{Comparison to the built-in \class{set} types
            \label{comparison-to-builtin-set}}

The built-in \class{set} and \class{frozenset} types were designed based
on lessons learned from the \module{sets} module.  The key differences are:

\begin{itemize}
\item \class{Set} and \class{ImmutableSet} were renamed to \class{set} and
      \class{frozenset}.
\item There is no equivalent to \class{BaseSet}.  Instead, use
      \code{isinstance(x, (set, frozenset))}.
\item The hash algorithm for the built-ins performs significantly better
      (fewer collisions) for most datasets.
\item The built-in versions have more space efficient pickles.
\item The built-in versions do not have a \method{union_update()} method.
      Instead, use the \method{update()} method which is equivalent.
\item The built-in versions do not have a \method{_repr(sorted=True)} method.
      Instead, use the built-in \function{repr()} and \function{sorted()}
      functions:  \code{repr(sorted(s))}.
\item The built-in version does not have a protocol for automatic conversion
      to immutable.  Many found this feature to be confusing and no one
      in the community reported having found real uses for it.
\end{itemize}