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A set is a Finite or Infinite collection of objects. Older words for set include Aggregate and Class. Russell also uses the term Manifold to refer to a set. The study of sets and their properties is the object of Set Theory. Symbols used to operate on sets include $\wedge$ (which denotes the Empty Set $\emptyset$), $\vee=$ (which denotes the Power Set of a set), $\cap$ (which means ``and'' or Intersection), and $\cup$ (which means ``or'' or Union).

The Notation $A^B$, where $A$ and $B$ are arbitrary sets, is used to denote the set of Maps from $B$ to $A$. For example, an element of $X^{\Bbb{N}}$ would be a Map from the Natural Numbers $\Bbb{N}$ to the set $X$. Call such a function $f$, then $f(1)$, $f(2)$, etc., are elements of $X$, so call them $x_1$, $x_2$, etc. This now looks like a Sequence of elements of $X$, so sequences are really just functions from $\Bbb{N}$ to $X$. This Notation is standard in mathematics and is frequently used in symbolic dynamics to denote sequence spaces.

Let $E$, $F$, and $G$ be sets. Then operation on these sets using the $\cap$ and $\cup$ operators is Commutative

E\cap F = F\cap E
\end{displaymath} (1)

E\cup F = F\cup E,
\end{displaymath} (2)

(E\cap F)\cap G = E\cap (F\cap G)
\end{displaymath} (3)

(E\cup F)\cup G = E\cup (F\cup G),
\end{displaymath} (4)

and Distributive
(E\cap F)\cup G = (E\cup G)\cap (F\cup G)
\end{displaymath} (5)

(E\cup F)\cap G = (E\cap G)\cup (F\cap G).
\end{displaymath} (6)

More generally, we have the infinite distributive laws
A\cap\left({\,\bigcup_{\lambda\in\Lambda} B_\lambda}\right)= \bigcup_{\lambda\in\Lambda} (A\cap B_\lambda)
\end{displaymath} (7)

A\cup\left({\,\bigcap_{\lambda\in\Lambda} B_\lambda}\right)= \bigcap_{\lambda\in\Lambda} (A\cup B_\lambda)
\end{displaymath} (8)

where $\lambda$ runs through any Index Set $\Lambda$. The proofs follow trivially from the definitions of union and intersection.

The table below gives symbols for some common sets in mathematics.

Symbol Set
$\Bbb{B}^n$ $n$-Ball
$\Bbb{C}$ Complex Numbers
$C^n$, $C^{(n)}$ $n$-Differentiable Functions
$\Bbb{D}^n$ $n$-Disk
$\Bbb{H}$ Quaternions
$\Bbb{I}$ Integers
$\Bbb{N}$ Natural Numbers
$\Bbb{Q}$ Rational Numbers
$\Bbb{R}^n$ Real Numbers in $n$-D
$\Bbb{S}^n$ $n$-Sphere
$\Bbb{Z}$ Integers
$\Bbb{Z}_n$ integers (mod $n$)
$\Bbb{Z}^-$ Negative Integers
$\Bbb{Z}^+$ Positive Integers
$\Bbb{Z}^*$ Nonnegative Integers

See also Aggregate, Analytic Set, Borel Set, C, Class (Set), Coanalytic Set, Definable Set, Derived Set, Double-Free Set, Extension, Ground Set, I, Intension, Intersection, Kinney's Set, Manifold, N, Perfect Set, Poset, Q, R, Set Difference, Set Theory, Triple-Free Set, Union, Venn Diagram, Well-Ordered Set, Z, Z-, Z+


Courant, R. and Robbins, H. ``The Algebra of Sets.'' Supplement to Ch. 2 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 108-116, 1996.

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© 1996-9 Eric W. Weisstein