Set

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

$$E\cap F = F\cap E$$ (1)

$$E\cup F = F\cup E,$$ (2)

Associative

$$(E\cap F)\cap G = E\cap (F\cap G)$$ (3)

$$(E\cup F)\cup G = E\cup (F\cup G),$$ (4)

and Distributive

$$(E\cap F)\cup G = (E\cup G)\cap (F\cup G)$$ (5)

$$(E\cup F)\cap G = (E\cap G)\cup (F\cap G).$$ (6)

More generally, we have the infinite distributive laws

$$A\cap\left({\,\bigcup_{\lambda\in\Lambda} B_\lambda}\right)= \bigcup_{\lambda\in\Lambda} (A\cap B_\lambda)$$ (7)

$$A\cup\left({\,\bigcap_{\lambda\in\Lambda} B_\lambda}\right)= \bigcap_{\lambda\in\Lambda} (A\cup B_\lambda)$$ (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+

References

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.