** Relations and Graphs**

The * cartesian product* of two sets
* A * and * B * , denoted
* A X B * , is the set of all ordered pairs * (a, b) * where
* a &isin A * and * b &isin B * .

A * relation* between two sets is a subset of their
cartesian product.

A * graph* is a pair * (S, &Gamma) * where * S * is
a set of nodes and * &Gamma &sube S X S * .

Properties of relations:

Property: | Definition: |

Reflexive | &forall a (a, a) &isin R |

Symmetric | &forall a, b (a, b) &isin R &rarr
(b, a) &isin R |

Transitive | &forall a, b, c (a, b) &isin R &and (b, c) &isin R |

&rarr (a, c) &isin R | |

Antisymmetric |
&forall a, b (a, b) &isin R &and (b, a) &isin R
&rarr a = b |

A relation that is reflexive, symmetric,
and transitive is an
* equivalence relation*, which corresponds to a
* partition*
of the set (a set of disjoint subsets whose union is the set).

A relation that is reflexive, antisymmetric, and transitive is a
* partial order*. Example: * &le * .