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sets.tex
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sets.tex
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%% Sets
SymPy supports representation of a wide variety of mathematical sets. This is
achieved by first defining abstract representations of atomic set classes and
then combining and transforming them using various set operations.
Each of the set classes inherits from the base class \texttt{Set} and defines
methods to check membership and calculate unions, intersections, and set
differences. When these methods are not able to evaluate to atomic set
classes, they are represented as abstract unevaluated objects.
SymPy has the following atomic set classes:
\begin{itemize}
\item \verb|EmptySet| represents the empty set $\emptyset$.
\item \verb|UniversalSet| is an abstract ``universal set'' of which
everything is a member. The union of the universal set with any set
gives the universal set and the intersection gives the other set itself.
\item \verb|FiniteSet| is functionally equivalent to Python's built
in \texttt{set} object. Its members can be any SymPy object including
other sets.
\item \verb|Integers| represents the set of integers $\mathbb{Z}$.
\item \verb|Naturals| represents the set of natural numbers $\mathbb{N}$,
i.e., the set of positive integers.
\item \verb|Naturals0| represents the set of whole numbers $\mathbb{N}_0$,
which are all the non-negative integers.
\item \verb|Range| represents a range of integers. A range is defined by
specifying a start value, an end value, and a step size. The enumeration
of a \texttt{Range} object is functionally equivalent to Python's
\texttt{range} except it supports infinite endpoints, allowing the
representation of infinite ranges.
\item \verb|Interval| represents an interval of real numbers. It is
defined by giving the start and the end points and by specifying if the interval is open
or closed on the respective ends.
\end{itemize}
%% Operations
Other than unevaluated classes of \texttt{Union}, \texttt{Intersection}, and
\texttt{Complement} operations, SymPy has the following set classes.
\begin{itemize}
\item \verb|ProductSet| defines the Cartesian product of two
or more sets. The product set is useful when representing higher
dimensional spaces. For example, to represent a three-dimensional space,
SymPy uses the Cartesian product of three real sets.
\item \verb|ImageSet| represents the image of a function when applied to
a particular set. The image set of a function $F$ with respect to a set
$S$ is $\{ F(x) \mid x \in S \}$. SymPy uses image sets to represent sets
of infinite solutions of equations such as $\sin(x)=0$.
\item \verb|ConditionSet| represents a subset of a set whose members
satisfy a particular condition. The subset of set $S$
given by the condition $H$ is $\{x \mid H(x), x \in S \}$. SymPy uses
condition sets to represent the set of solutions of equations and
inequalities, where the equation or the inequality is the condition and
the set is the domain over which it is being solved.
\end{itemize}
A few other classes are implemented as special cases of the classes described
above. The set of real numbers, \verb|Reals|, is implemented as a special case
of \verb|Interval|. \verb|ComplexRegion|
is implemented as a special case of \verb|ImageSet|. \verb|ComplexRegion|
supports both polar and rectangular representation of regions on the complex
plane.