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Mathematics-Online lexicon:

Set

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A set $ A$ consists of distinct elements $ a_1,a_2,\ldots$:

$\displaystyle A = \{a_1,a_2,\ldots\}\, . $

If such elements are characterized via a property $ E$, this is symbolized as follows:

$\displaystyle A = \{a:\ a\ $   satisfies property$\displaystyle \ E\}\, . $

The following notations are commonly used:

notation meaning
$ a\in A$ $ a$ is element of $ A$
$ a\notin A$ $ a$ is not element of $ A$
$ A\subseteq B$ $ A$ is a subset of $ B$
$ A\subset B$ $ A$ is a strict subset of $ B$
$ \vert A\vert$ number of elements in $ A$
$ \emptyset$ empty set

If $ \vert A\vert<\infty$ ($ =\infty$), $ A$ is called a finite (infinite) set.

Two sets are called equipotent, if there exists a bijective map between their elements ($ \vert A\vert=\vert B\vert$ for finite sets $ A$ and $ B$).

The set $ {\cal P}(A)$ of all subsets of $ A$ is called power set, i.e. $ {\cal P}(A)=\{B:\ B\subseteq A\}$. In particular, we have $ \emptyset\in {\cal P}(A)$ and $ A\in {\cal P}(A)$. Moreover, $ \vert{\cal P}(A)\vert=2^{\vert A\vert}$.

see also:

[Examples]
  automatically generated 5/18/2009