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 (which denotes the Empty Set ), (which denotes the Power Set of a set), (which means ``and'' or Intersection), and (which means ``or'' or Union).

The Notation , where and are arbitrary sets, is used to denote the set of Maps from to . For example, an element of would be a Map from the Natural Numbers to the set . Call such a function , then , , etc., are elements of , so call them , , etc. This now looks like a Sequence of elements of , so sequences are really just functions from to . This Notation is standard in mathematics and is frequently used in symbolic dynamics to denote sequence spaces.

Let , , and be sets. Then operation on these sets using the and operators is Commutative

(1) |

(2) |

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

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

Symbol | Set |

-Ball | |

Complex Numbers | |

, | -Differentiable Functions |

-Disk | |

Quaternions | |

Integers | |

Natural Numbers | |

Rational Numbers | |

Real Numbers in -D | |

-Sphere | |

Integers | |

integers (mod ) | |

Negative Integers | |

Positive Integers | |

Nonnegative Integers |

**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.

© 1996-9

1999-05-26